For a projective variety $X$ and integers $g, n\geq 0$ Kontsevich introduced moduli spaces $K_{g, n}(X)$ classifying stable maps from $n$-marked genus $g$ curves to $X$.  There are two natural generalizations of this theory.  The first, due in the algebraic context to Abramovich and Vistoli, is to allow the target $X$ to be a stack, which is natural from the point of view of classifying families of geometric objects over curves.  The second, due in various forms to Hassett Alexeev and Guy, and Bayer and Manin, is to allow fractional coefficients of the marked points on the curves.  This is natural from the point of view of the minimal model program.  In this talk, I will discuss moduli spaces of weighted stable maps to stacks, which is a common generalization of all of these approaches. 

This is joint work with Rachel Webb.