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Since nite simple groups are the building blocks of nite groups, it is natural to ask about their occurrence \in nature". In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of nite subgroups of algebraic groups. summary. The classication program for nite subgroups of complex algebraic groups involves both existence of embeddings and their classication up to conjugacy. We have just classied embeddings of Sz(8) into E8(C) (there are three, up to E8(C)- conjugacy), thus settling the existence question for projective embeddings of nite simple groups into exceptional algebraic groups. The conjugacy part of the program is only partially resolved. The nite subgroups of the smallest simple algebraic group PSL (2;C )( up to conjugacy) constitute the famous list: cyclic, dihedral, Alt4, Sym4, Alt5. This list has been associated to geometry, number theory, and Lie theory in several ways. McKay's correspondence between these groups and the Cartan matrices of types A, D and E and his related tensor product observations (McK) are provocative. For the exceptional algebraic groups, theories of Kostant, Springer and Serre have called attention to particular nite simple subgroups. A good list of nite subgroups should help us understand the exceptional groups better. Table of contents

Finite simple groups which projectively embed in an exceptional Lie group are classified!