"k-matching" in graphs A matching in a graph is a set of edges that are pair-wise non-adjacent. IOW, each node involved in the matching appears in only one edge.  

Now I am wondering is there a ``generalized'' concept of matching, such that it still refers to an edge-set, and each vertex involved can appear in at most $k$ edges in this edge-set, where $k \ge 2$ is a pre-specified positive integer.

If such concept does exist, what are some interesting and representative problems studied for that?
A short answer is http://en.wikipedia.org/wiki/Graph_factorization.  A more comprehensive reference might be the classic "Matching Theory" by Lovasz and Plummer, which is the best on this topic.