  Heegaard Floer theory is a kind of topological quantum field theory,
assigning graded groups to closed, connected, oriented 3-manifolds and group
homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard
Floer homology is an extension of Heegaard Floer homology to 3-manifolds with
boundary, with extended-TQFT-type gluing properties. In this survey, we explain
the formal structure and construction of bordered Floer homology and sketch how
it can be used to compute some aspects of Heegaard Floer theory.
