Abstract:  We will discuss two different compactifications of the moduli space of maps from smooth curves to punctorial Hilbert schemes of a surface. Namely, the moduli space of stable maps (Gromov-Witten theory) and the relative Hilbert scheme of one-dimensional subschemes (Donaldson-Thomas theory) on threefolds of the type Surface x Curve for a varying nodal curve. The (virtual) intersection theories of these moduli spaces are related by a certain wall-crossing, which is provided by the theory of quasimaps to moduli spaces of sheaves.