  The Minimal Model Program offers natural higher-dimensional analogues of
stable $n$-pointed curves and maps: stable pairs consisting of a projective
variety $X$ of dimension $\ge2$ and a divisor $B$, that should satisfy a few
simple conditions, and stable maps $f:(X,B)\to Y$. Although MMP remains
conjectural in higher dimensions, in several important situations the moduli
spaces of stable pairs, generalizing those of Deligne-Mumford, Knudsen and
Kontsevich, can be constructed more directly, and in considerable generality.
We review these constructions, with particular attention paid to varieties with
group action, and list some open problems.
