I will first discuss some fairly classical homological stability phenomena: spaces of 0-manifolds (e.g. configuration spaces), and spaces of 2-manifolds (e.g. Riemann's moduli space). I will explain the general method, introduced by Quillen, for proving these stability theorems, and then explain some recent work with Søren Galatius which proves such a stability theorem for moduli spaces of 2n-dimensional manifolds (for n > 2).