The Steinberg relations are the commutator relations which hold between elementary matrices in a special linear group. This text generalizes these sorts of relations. To encode these relations one needs a ring and a so-called linkage graph which specifies exactly which commutator relations hold. The groups obtained here, called linkage groups, have an enormous number of interesting images, finite and infinite. Among these images are, for example, 25 of the 26 finite sporadic simple groups. The book deals with the structure and classification of linkage groups. Part of the work involves theoretical group combinatorics and the other part involves computer calculations to study the linkage structure of various interesting groups. The book will be of value to researchers and graduate students in combinatorial and computational group theory.