Chern-Weil theory describes a procedure for constructing the characteristic classes of a smooth manifold from geometric data (such as a Riemannian metric). In the 1970s and 1980s, Chern-Weil theory was successfully adapted by R. Bott to describe the characteristic classes of the leaf space of any regular foliation, including the so-called secondary classes such as the Godbillon-Vey invariant. As discovered by Connes and Moscovici, these characteristic classes also play a key role in noncommutative geometry, where they define cyclic cocycles that pair with the K-theory of the noncommutative leaf space. The extension of Chern-Weil theory to singular foliations has, however, remained elusive. In this talk, I will describe recent, joint work with Benjamin McMillan which gives a Chern-Weil homomorphism for a family of singular foliations whose singularities are not “too big”.