Complexes of groups G(X) over ordered simplicial complexes X are generalizations to higher dimensions of graphs of groups. We first relate them to complexes of spaces by considering their classifying space BG(X). Then we develop their homological algebra aspects. We define the notions of homology and cohomology of a complex of groups G(X) with coefficients in a G(X)-module and show the existence of free resolutions. We apply those notions to study extensions of complexes of groups with constant or abelian kernel.