The paper was prepared to assist the American Bar Association's Task Force studying Federal habeas corpus review of State death penalty convictions. The overall aim of the task-force project is to develop recommendations that will produce efficient and fair review procedures in death penalty cases. The paper addresses the purpose of postconviction review of State criminal convictions; the competence, provision, and zeal of counsel; and State procedural default rules. Also covered are the exhaustion of State judicial remedies; successive petitions, abuse of the writ, and delay; and certificates of probable cause, stays of execution, and last-minute "chaos" in the review process. The paper suggests that an effective solution to the problems of last-minute Federal court review of capital cases and requests for stays of execution will be found by investigating the needs of the entire criminal justice processing of capital cases, rather than in a focus on the last stage of the review process. 470 footnotes, 75-item bibliography, appended habeas corpus statutes and rules.