If K is a non-abelian group on which a group G acts via automorphisms, we can define 1-cocycles and 1-coboundaries by mimicking the explicit formulas coming from the bar resolution in ordinary group cohomology, and thus we have a reasonable notion of H^1(G, K). 

It turns out we have a part of the expected long exact sequence, until this construction breaks down for building H^i when i > 1, where the long exact sequence stops. There are other analogues to ordinary group cohomology as well. The only proof I've ever seen of any of this is by hand. Is there some deeper explanation of why non-abelian group cohomology exists (and then ceases to exist)?