delivered the opinion of the Court.
On May 4, 1982, after eating their evening meal in their Muskogee County, Oklahoma, home, Hugh and Charma Riddle watched television in their living room. At some point, Mrs. Riddle left the living room and was proceeding towards the bathroom when she encountered respondent Cartwright standing in the hall holding a shotgun. She struggled for the gun and was shot twice in the legs. The man, whom she recognized as a disgruntled ex-employee, then proceeded to the living room where he shot and killed Hugh Riddle. Mrs. Riddle dragged herself down the hall to a bedroom where she tried to use a telephone. Respondent, however, entered the bedroom, slit Mrs. Riddle’s throat, stabbed her twice with a hunting knife the Riddles had given him for Christmas, and then left the house. Mrs. Riddle survived and called the police. Respondent was arrested two days later and charged with first-degree murder.
Respondent was tried and found guilty as charged. The State, relying on three statutory aggravating circumstances, sought the death penalty. The jury found two of them to have been established: first, the defendant “knowingly cre*359ated a great risk of death to more than one person”; second, the murder was “especially heinous, atrocious, or cruel.” Okla. Stat., Tit. 21, §§ 701.12(2) and (4) (1981). Finding that the aggravating circumstances outweighed the mitigating evidence, the jury imposed the death penalty. The Oklahoma Court of Criminal Appeals affirmed on direct appeal, Cartwright v. State, 695 P. 2d 548, cert. denied, 473 U. S. 911 (1985), and later affirmed a denial of state collateral relief. Cartwright v. State, 708 P. 2d 592 (1985), cert. denied, 474 U. S. 1073 (1986). Respondent then sought federal habeas corpus on several grounds. The District Court rejected each of them, including the claim that the death sentence was invalid because it rested wholly or in part on an unconstitutional aggravating .circumstance, namely, the unconstitutionally vague and overbroad aggravating circumstance that the murder was “especially heinous, atrocious, or cruel.” A panel of the Court of Appeals for the Tenth Circuit affirmed, 802 F. 2d 1203 (1986), but rehearing en banc was granted limited to the claim concerning the challenged aggravating circumstance.
The en banc court recognized that the jury had found two aggravating circumstances, one of them being unchallenged. But it noted that in cases where a death sentence rested in part on an invalid aggravating circumstance, the Oklahoma courts did not reweigh the aggravating and mitigating circumstances in an effort to save the death penalty; rather, the death sentence was vacated and a life-imprisonment sentence automatically imposed. Oklahoma had “no provision fdr curing on appeal a sentencer’s consideration of an invalid aggravating circumstance.” 822 F. 2d 1477, 1482 (1987). It was therefore necessary to consider the vagueness challenge to one of the aggravating circumstances. The court proceeded to do so and unanimously sustained the challenge. It stated that the words “heinous,” “atrocious,” and “cruel” did not on their face offer sufficient guidance to the jury to escape the strictures of our judgment in Furman v. Georgia, *360408 U. S. 238 (1972). Nor, in the court’s view, had the Oklahoma courts adopted a limiting construction that cured the infirmity and that was relied upon to affirm the death sentence in this case. It concluded that the Oklahoma Court of Criminal Appeals’ construction of the aggravating circumstance was “unconstitutionally vague” under the Eighth Amendment. 822 F. 2d, at 1483, 1492. The death sentence, accordingly, was held to be invalid and its execution enjoined, but “without prejudice to further proceedings by the state for redetermination of the sentence on the conviction.” Id., at 1492.
Petitioner sought review here of the Tenth Circuit’s holding that the aggravating circumstance was unconstitutionally vague. Because of the conflict between the Court of Appeals for the Tenth Circuit and the Court of Criminal Appeals of Oklahoma and because of the importance of this constitutional issue to the orderly and proper administration of state death-penalty statutes, we granted certiorari, limited to that issue. 484 U. S. 1003 (1988). We affirm the judgment of the Court of Appeals.
The Court of Appeals, with some care, reviewed the evolution in the interpretation of the “especially heinous, atrocious, or cruel” aggravating circumstance by the Oklahoma Court of Criminal Appeals up to and including its decision in this case. Its reading of the cases was that while the Oklahoma court had considered the attitude of the killer, the manner of the killing, and the suffering of the victim to be relevant and sufficient to support the aggravating circumstance, that court had “refused to hold that any one of those factors must be present for a murder to satisfy this aggravating circumstance.” 822 F. 2d, at 1491. Rather, the Oklahoma court simply had reviewed all of the circumstances of the murder and decided whether the facts made opt the aggravating circumstance. Ibid. We normally defer to courts of appeals in their interpretation of state law, and we see no reason not to accept the Court of Appeals’ statements about *361state law in this case, especially since the State does not challenge this reading of the Oklahoma cases.
The State, however, takes issue with the Court of Appeals’ conclusion that this approach, which was also employed in this case, to interpreting and applying the challenged aggravating circumstance is unconstitutional. It insists that in some cases there are factual circumstances that so plainly characterize the killing as “especially heinous, atrocious, or cruel” that affirmance of the death penalty is proper. As we understand the argument, it is that a statutory provision governing a criminal case is unconstitutionally vague only if there are no circumstances that could be said with reasonable certainty to fall within reach of the language at issue. Or to put it another way, that if there are circumstances that any reasonable person would recognize as covered by the statute, it is not unconstitutionally vague even if the language would fail to give adequate notice that it covered other circumstances as well.
The difficulty with the State’s argument is that it presents a Due Process Clause approach to vagueness and fails to recognize the rationale of our cases construing and applying the Eighth Amendment. Objections to vagueness under the Due Process Clause rest on the lack of notice, and hence may be overcome in any specific case where reasonable persons would know that their conduct is at risk. Vagueness challenges to statutes not threatening First Amendment interests are examined in light of the facts of the case at hand; the statute is judged on an as-applied basis. United States v. Powell, 423 U. S. 87, 92-93 (1975); United States v. Mazurie, 419 U. S. 544, 550 (1975); Palmer v. City of Euclid, 402 U. S. 544 (1971) (per curiam); United States v. National Dairy Corp., 372 U. S. 29, 32-33, 36 (1963). Claims of vagueness directed at aggravating circumstances defined in capital punishment statutes are analyzed under the Eighth Amendment and characteristically assert that the challenged provision fails adequately to inform juries what they must *362find to impose the death penalty and as a result leaves them and appellate courts with the kind of open-ended discretion which was held invalid in Furman v. Georgia, 408 U. S. 238 (1972).
Furman held that Georgia’s then-standardless capital punishment statute was being applied in an arbitrary and capricious manner; there was no principled means provided to distinguish those that received the penalty from those that did not. E. g., id., at 310 (Stewart, J., concurring); id., at 311 (White, J., concurring). Since Furman, our cases have insisted that the channeling and limiting of the sentencer’s discretion in imposing the death penalty is a fundamental constitutional requirement for sufficiently minimizing the risk of wholly arbitrary and capricious action. Gregg v. Georgia, 428 U. S. 153, 189, 206-207 (1976) (opinion of Stewart, Powell, and Stevens, JJ.); id., at 220-222 (White, J., concurring in judgment); Spaziano v. Florida, 468 U. S. 447, 462 (1984); Lowenfield v. Phelps, 484 U. S. 231, 244 (1988).
Godfrey v. Georgia, 446 U. S. 420 (1980), which is very relevant here, applied this central tenet of Eighth Amendment law. The aggravating circumstance at issue there permitted a person to be sentenced to death if the offense “was outrageously or wantonly vile, horrible or inhuman in that it involved torture, depravity of mind, or an aggravated battery to the victim.” Id., at 422. The jury had been instructed in the words of the statute, but its verdict recited only that the murder was “outrageously or wantonly vile, horrible or inhuman.” The Supreme Court of Georgia, in affirming the death sentence, held only that the language used by the jury was “not objectionable” and that the evidence supported the finding of the presence of the aggravating circumstance, thus failing to rule whether, on the facts, the offense involved torture or an aggravated battery to the victim. Id., at 426-427. Although the Georgia Supreme Court in other cases had spoken in terms of the presence or absence of these factors, it did not do so in the decision under review, *363and this Court held that such an application of the aggravating circumstance was unconstitutional, saying:
“In the case before us, the Georgia Supreme Court has affirmed a sentence of death based upon no more than a finding that the offense was ‘outrageously or wantonly vile, horrible and inhuman.’ There is nothing in these few words, standing alone, that implies any inherent restraint on the arbitrary and capricious infliction of the death sentence. A person of ordinary sensibility could fairly characterize almost every murder as ‘outrageously or wantonly vile, horrible and inhuman.’ Such a view may, in fact, have been one to which the members of the jury in this case subscribed. If so, their preconceptions were not dispelled by the trial judge’s sentencing instructions. These gave the jury no guidance concerning the meaning of any of [the aggravating circumstance’s] terms. In fact, the jury’s interpretation of [that circumstance] can only be the subject of sheer speculation.” Id., at 428-429 (footnote omitted).
The affirmance of the death sentence by the Georgia Supreme Court was held to be insufficient to cure the jury’s unchanneled discretion because that court failed to apply its previously recognized limiting construction of the aggravating circumstance. Id., at 429, 432. This Court concluded that, as a result of the vague construction applied, there was “no principled way to distinguish this case, in which the death penalty was imposed, from the many cases in which it was not.” Id., at 433. Cf. Proffitt v. Florida, 428 U. S. 242, 254-256 (1976). It plainly rejected the submission that a particular set of facts surrounding a murder, however shocking they might be, were enough in themselves, and without some narrowing principle to apply to those facts, to warrant the imposition of the death penalty.
We think the Court of Appeals was quite right in holding that Godfrey controls this case. First, the language of the Oklahoma aggravating circumstance at issue — “especially *364heinous, atrocious, or cruel” — gave no more guidance than the “outrageously or wantonly vile, horrible or inhuman” language that the jury returned in its verdict in Godfrey. The State’s contention that the addition of the word “especially” somehow guides the jury’s discretion, even if the term “heinous” does not, is untenable. To say that something is “especially heinous” merely suggests that the individual jurors should determine that the murder is more than just “heinous,” whatever that means, and an ordinary person could honestly believe that every unjustified, intentional taking of human life is “especially heinous.” Godfrey, supra, at 428-429. Likewise, in Godfrey the addition of “outrageously or wantonly” to the term “vile” did not limit the overbreadth of the aggravating factor.
Second, the conclusion of the Oklahoma court that the events recited by it “adequately supported the jury’s finding” was indistinguishable from the action of the Georgia court in Godfrey, which failed to cure the unfettered discretion of the jury and to satisfy the commands of the Eighth Amendment. The Oklahoma court relied on the facts that Cartwright had a motive of getting even with the victims, that he lay in wait for them, that the murder victim heard the blast that wounded his wife, that he again brutally attacked the surviving wife, that he attempted to conceal his deeds, and that he attempted to steal the victims’ belongings. 695 P. 2d, at 554. Its conclusion that on these facts the jury’s verdict that the murder was especially heinous, atrocious, or cruel was supportable did not cure the constitutional infirmity of the aggravating circumstance.
The State complains, however, that the Court of Appeals ruled that to be valid the “especially heinous, atrocious, or cruel” aggravating circumstance must be construed to require torture or serious physical abuse and that this was error. We do not, however, agree that the Court of Appeals imposed this requirement. It noted cases in which such a requirement sufficed to validate an otherwise vague aggravat*365ing circumstance, but it expressly refrained from directing the State to adopt any specific curative construction of the aggravating circumstance at issue here. 822 F. 2d, at 1491-1492. We also do not hold that some kind of torture or serious physical abuse is the only limiting construction of the heinous, atrocious, or cruel aggravating circumstance that would be constitutionally acceptable.
The State also insists that the death penalty should stand because the jury found two aggravating circumstances, one of which was unchallenged and is sufficient to sustain the sentence. When this case was decided, however, the Oklahoma Court of Criminal Appeals would not attempt to save the death penalty when one of several aggravating circumstances found by the jury was found invalid or unsupported by the evidence. As the Tenth Circuit said, there was “no provision for curing on appeal a sentencer’s consideration of an invalid aggravating circumstance.” Id., at 1482. If this was the case at that time, and the State does not dispute it, the Court of Appeals cannot be faulted for not itself undertaking what the state courts themselves refused to do.
It is true that since the decision of the Court of Appeals, the Oklahoma Court of Criminal Appeals has restricted the “heinous, atrocious, or cruel” aggravating circumstance to those murders in which torture or serious physical abuse is present. Stouffer v. State, 742 P. 2d 562 (1987). At the same time, that court decided that it would not necessarily set aside a death penalty where on appeal one of several aggravating circumstances has been found invalid or unsupported by the evidence. Id., at 564. See also Castro v. State, 745 P. 2d 394, 408-409 (1987), cert. denied, 485 U. S. 971 (1988).
What significance these decisions of the Court of Criminal Appeals have for the present case is a matter for the state courts to decide in the first instance. Like that of the Court of Appeals, our judgment is without prejudice to further pro*366ceedings in the state courts for redetermination of the appropriate sentence.
The judgment of the Court of Appeals is
Affirmed.