Chain complexes are most often encountered in an Abelian context, but the definition makes sense even when the groups involved need not be abelian. What does not work well is the formation of the homology of such a chain complex of groups, except if it is ‘normal’ in the following sense:

Definition

A chain complex of groups is a sequence (of any length, finite or infinite) of groups and homomorphisms, for instance,

There is an obvious generalisation to normal complex of groupoids, provided one keeps to working with groupoids all having the same set of objects and morphisms which are the identity on objects.