Stable maps

Let X be a projective variety. A stable map to X is a map f:C -> X, where C is a nodal connected marked (prestable) curve, and only finitely many automorphisms of C commute with f. We say that f is a genus-g, n-marked stable map of degree d if C has genus g and n marked points, and f_*[C]=d.

Moduli
For any choice of g, n, and d (other than (0,0,0), (0,1,0), (0,2,0), (1,0,0)), the moduli stack Mgn(X,d) is a proper Deligne-Mumford stack (with a canonical perfect obstruction theory and virtual fundamental class).