It is commonly stated as:

In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, it is always possible to solve the SMP and make all marriages stable. They presented an algorithm to do so.

The Gale-Shapley algorithm involves a number of "rounds" (or "iterations") where each unengaged man "proposes" to the most-preferred woman to whom he has not yet proposed. Each woman then considers all her suitors and tells the one she most prefers "Maybe" and all the rest of them "No". She is then provisionally "engaged". In each subsequent round, each unengaged man proposes to one woman to whom he has not yet proposed (the woman may or may not already be engaged), and the women once again reply with one "maybe" and reject the rest. This may mean that already-engaged women can "trade up", and already-engaged men can be "jilted".

This algorithm guarantees that:

The pairing generated by the Gale-Shapley algorithm has a number of interesting properties. In particular, one may prove that, in some sense, the Gale-Shapley pairing, in the form presented here, is male-optimal and female-pessimal (it would be the reverse, of course, if the roles of "male" and "female" participants in the algorithm were interchanged). To see this, consider the definition of a feasible marriage. We say that the marriage between man A and woman B is feasible if there exists a stable pairing in which A and B are married. When we say a pairing is male-optimal, we mean that every man is paired with his highest ranked feasible partner. Similarly, a female-pessimal pairing is one in which each female is paired with her lowest ranked feasible partner.

Proof that the Gale-Shapley pairing is male-optimal: We go by contradiction. Suppose the pairing generated by the Gale-Shapley algorithm were not male-optimal. Let T be the first iteration where a man is rejected by his optimal partner. Let man M be one man who is rejected by his optimal partner, woman W, during this iteration. Then woman W must have rejected man M for some other man, man Z. Further, since this is the first iteration where a man is rejected by his optimal partner, woman W must be at least as high on man Z's preference list as his optimal partner. Also note that, since woman W is man M's optimal partner, there exists some stable pairing S where man M is paired with woman W. Then, in pairing S, woman W prefers man Z to man M (since she rejected M for Z in the Gale-Shapley algorithm) and man Z prefers woman W to his partner in S, since he likes woman W at least as much as his optimal partner and, by our definition of optimal, he could not be paired with anyone better than this. But then, by definition, this pairing is unstable, since man Z and woman W prefer each other to their partners in S. We have reached a contradiction, so the Gale-Shapley algorithm must produce a male-optimal pairing.

Proof that the Gale-Shapley pairing is female-pessimal: We can show this using the fact a male-optimal pairing must be female-pessimal. Consider any woman W paired to a man M in the male-optimal stable pairing G. Also consider an arbitrary stable pairing S, where M and W are not paired. Because G is male-optimal, M prefers W to his match in S. Because S is stable, W must prefer her match in S to M. Thus women will always prefer another match in any stable pairing over their match in a male-optimal pairing. Thus, the Gale-Shapley pairing is female-pessimal because it is male-optimal (see above).

There are a number of real world applications for the Gale-Shapley solution. For example, their algorithm has been used to pair graduating medical students with hospital jobs.

The weighted matching problem seeks to find a matching in a weighted bipartite graph that has maximum weight. Maximum weighted matchings do not have to be stable, but in some applications a maximum weighted matching is better than a stable one.

The stable roommates problem is similar to the stable marriage problem, but differs in that all participants belong to a single pool (instead of being divided into equal numbers of "men" and "women").