Groups comprising two subcomponents are of particular interest to group theorists who want to know in what way the structure of the product is related to that of its subgroups. This monograph gives the first detailed account of the most important results of group product research from the past 35 years. Although the emphasis is on infinite groups, relevant theorems about finite products of groups are also proved. This book will be of interest to research students and specialists in group theory, and will be useful in seminars or as a supplement in courses in general group theory. A special chapter on conjugacy and splitting theorems obtained by means of the cohomology of groups has never before appeared in book form.