摘要： We recall how the Jones polynomial of links lifts to a link homology theory which is functorial under link cobordisms in 4-space. Extending this homology theory to tangles relates the theory to a variety of categories in representation theory and geometry. Another extension, to a categorification of the Reshetikhin-Turaev invariants for SL(N), requires working with foams as an intermediate step. We will review these developments and sketch some open problems in the theory.