Synopsis The theory of infinite permutation groups is a branch of algebra which has developed rapidly during the last twenty years, driven partly by connections with mathematical logic. This book is based on a course of lectures at the Indian Institute of Technology, Guwahati, in August and September 1996. It takes the reader from the most basic notions in group theory to a recently completed classification result for Jordon groups. Topics examined include : transitivity and primitivity; symmetric groups; general linear groups; wreath products; automorphism groups of various treelike objects; some model-theoretic techniques for building structures with rich automorphism groups; the structure and classification of infinite primitive Jordon groups; applications and open problems. There are examples and exercises throughout the text.