Khovanov homology, 3-manifolds, and 4-manifolds

Khovanov homology is an invariant of knots in R^3. Two open problems are to extend its definition to knots in other three-manifolds, and to understand its relation to surfaces in 4-manifolds. I will discuss some partial progress in these directions. In the process I will also review some of the topological applications of Khovanov homology.