Part of the difficulty of finding out more about this problem is that the graph matching problem is different from its much more famous cousin, the matching problem, but hard to be distinguished from it when using search engines.
Given two graphs $G=(V,E)$ and $G'=(V',E')$ such that $|V| = |V'|$, the task is to find a bijection $\pi : V \rightarrow V'$ such that this bijection establishes as much correspondences between edges of $G$ and $G'$ as possible.
In other words, if $M$ and $M'$ are the adjascency matrices, then we want to maximize
$\sum_{v,w \in V} M_{v,w} \cdot M'_{\pi(v),\pi(w)}$
This problem clearly contains graph isomorphism as a special case, and can be reduced to bipartite matching under a (non-polynomial!) reduction.
What kind of algorithms do exist, and what is known about its complexity?