The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \mathbb R^n. By classical results of Caffarelli, the free boundary is C^\infty outside a set of singular points. Explicit examples show that the singular set could be in general (n−1)-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero \mathcal H^{n-4} measure (in particular, it has codimension 3 inside the free boundary). In particular, for n\leq4, the free boundary is generically a C^\infty manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n\leq4
