The low-dimensional topology community has been energized in recent years by the introduction of a wealth of so-called ``homology-type" invariants. One associates to an object in low-dimensional topology (e.g., a link or a 3-manifold) an abstract chain complex whose homology is an invariant of the topological object. Such invariants arise in two apparently different ways: ``algebraically," via the representation theory of quantum groups and ``geometrically," via constructions in symplectic geometry. I will discuss what is known about the relationship between two such invariants: Khovanov homology, an ``algebraic" invariant of links and tangles defined by Khovanov and Heegaard-Floer homology, a ``geometric" invariant of 3-manifolds defined by Ozsvath-Szabo. The portions of the talk describing my own work are joint with Denis Auroux and Stephan Wehrli.