The current constitutional requirements for the imposition of a death sentence are presented, and the principles essential to them are developed. The constitutional status of judicial review of death sentences is then considered, and it is concluded that forms of judicial review which reduce arbitrariness are constitutionally mandated. The concluding section discusses problems in judicial review of death sentences and describes constitutionally adequate systems of capital sentence review. It is concluded that capital sentencing continues to be arbitrary in constitutionally objectionable ways. Rigorous techniques of appellate and proportionality review are advocated. Proportionality review also ensures that capital sentencing will reflect current community mores regarding the death sentence and helps to guarantee equal protection for capital defendants. For these reasons, the United States Supreme Court should hold that rigorous appellate review and particularly proportionality review are constitutionally required. Those who assert that rigorous appellate and proportionality review are not constitutionally required must claim either that there is no arbitrariness in State court decisions or that an arbitrary imposition of death is proper. Appellate review of capital sentencing has been demonstrably deficient. Courts have been highly subjective in their case characterizations and have resisted efforts to objectify the case comparison process. Only a change in procedural requirements, imposing an obligation on the sentencer to give reasons for the sentence, can resolve the problem. Existing quantitative methods offer promise. Data for each capital case can be collected on all significant factors. Through statistical analysis of the cases a person can determine which factors or which combination of factors appear to explain the verdicts. Combined with qualitative review, quantitative methods would assist a reviewing court in focusing on significant sentencing factors. A total of 193 footnotes are provided.