I discuss the general formalism of two-dimensional topological field theories
defined on open-closed oriented Riemann surfaces, starting from an extension of
Segal's geometric axioms. Exploiting the topological sewing constraints allows
for the identification of the algebraic structure governing such systems. I
give a careful treatment of bulk-boundary and boundary-bulk correspondences,
which are responsible for the relation between the closed and open sectors. The
fact that these correspondences need not be injective nor surjective has
interesting implications for the problem of classifying `boundary conditions'.
In particular, I give a clear geometric derivation of the (topological)
boundary state formalism and point out some of its limitations. Finally, I
formulate the problem of classifying (on-shell) boundary extensions of a given
closed topological field theory in purely algebraic terms and discuss their
reducibility.