cdh methods in K-theory and Hochschild homology

This is intended to be a survey talk, accessible to a general mathematical audience. The cdh topology was created by Voevodsky to extend motivic cohomology from smooth varieties to singular varieties, assuming resolution of singularities (for example complex varieties). The slogan is that every variety is locally smooth in this topology. Adapting this to algebraic K-theory has led to several breakthroughs in computation, and clarified connections between the "singular" part of K-theory and Hodge theory.