On the preimage of maximal ideals

Question

Let $\varphi:R\to S$ be a map of commutative rings. It is well known that if $M$ is maximal in $S$ then the preimage $\varphi^{-1}(M)$ need not be maximal in $R$ (consider $\mathbb{Z}\to\mathbb{Q}$). There are however special cases such as the following:

1. If $\varphi$ is surjective, then $\varphi^{-1}(M)$ is maximal in $R$.
2. If $R$ and $S$ are algebras over a field $K$, and $S$ is finitely generated and $\varphi$ is a map of $K$-algebras, then $\varphi^{-1}(M)$ is maximal in $R$.

Question: How much can be said about the converse? Namely, if $\varphi$ satisfies the property that $\varphi^{-1}(M)$ is maximal in $R$ for every maximal in $S$, what can be said about $R$, $S$ and $\varphi$? I am assuming not much, so can we make additional assumptions to make this question interesting? Any references are welcome.