Have any generalizations of maximum weight matching been studied?
There are several extensions of the problem to more general structures.  For instance:


Matroid matching (lecture notes, Matroid matching and some applications, Matroid Matching: the Power of Local Search)
Path-matching (Path-matching problems, Matching, matroids, and extensions)
Hypergraph matching (Can't find any good references)


Generally, these extensions are NP-hard.