This monograph focuses on progress in the study of Sylow subgroups and their influence on the structure of the group as a whole. This research has been applied to other areas of finite group theory, including classification of simple groups, but is also of independent interest and does not require extensive background or long proofs. In 1969, the author gave a report on this topic which appeared in the book ""Finite Simple Groups"" (edited by M. B. Powell and G. Higman, Academic Press, 1971; MR 48 No. 6228).The present monograph covers progress since 1969. It includes some new results of Yoshida on transfer, a partial analogue for $p-2$ of the author's ""$ZJ$-Theorem"", and the classification of all simple groups which are $S^4$-free, i.e., in which the symmetric group of degree four is not involved. It also includes an expository account of recent work of M. Aschbacher, B. Baumann, R. Niles, and others on 'failure of factorization', 'pushing-up' arguments, and related subjects. This is not an expository work. This work should be accessible to advanced graduate students. In particular, a semester's study in finite group theory beyond the M.A. or M.S. degree should be adequate background, e. g., Chapters 1-3 and 5-7 of Gorenstein's Reviews on finite groups (Amer. Math. Soc., 1974; MR 50 < No. 2312). The book supplements the author's report in Finite simple groups. Familarity with this report is recommended but not assumed.
