Let S be a smooth projective surface. The Hilbert schemes Hilb^n(S) of n points on S are well-understood and central objects in geometry. Less is known, however, about degenerations of such varieties. In this talk, I will present joint work with M. Gulbrandsen, K. Hulek and Z. Zhang, where we study how the Hilbert scheme degenerates "along" with the surface S. I will also discuss a few applications of our construction, and some open problems.