What is a non-bipartite matching problem and what are some of the real life applications of non-bipartite matching?

I know what bipartite matching is. In bipartite matching, we look for a one-to-one match between two disjoint sets of vertices. A classic real-life example of bipartite matching is matching kidneys. I am wondering what non-bipartite matching is really about and where we may want to solve non-bipartite matching problems in real life.

1. Does non-bipartite matching refers to cases where we have more than two disjoint sets of vertices (such as tripartite)?
2. Does non-bipartite matching mean that we have a bipartite network, but we can also have matching within a specific set?
1. and 2. both can happen?
• I would think a bipartite matching would be a matching of items of one type with items of a different type, whereas a non-bipartite matching would be a matching of items of one type with other items of the same type (that is, a matching of items in the same set). Commented Nov 10, 2022 at 6:36
• @GerryMyerson, I think your comment refers to the second case that I mentioned in my question. Can we have both within the same set and between two different set matchings at the same time? Commented Nov 10, 2022 at 6:58
• I don't know. It seems to me that either you have a matching between set $A$ and set $B$, or else you have a matching on a set $A$. I suppose you could have a matching where some elements of $A$ are matched with elements of $B$, and some elements of $A$ (respectively, $B$) are matched with elements of $A$ (respectively, $B$). I wouldn't call that bipartite. Commented Nov 10, 2022 at 10:28