  We study a special sort of 2-dimensional extended Topological Quantum Field
Theories (TQFTs) which we call open-closed TQFTs. These are defined on
open-closed cobordisms by which we mean smooth compact oriented 2-manifolds
with corners that have a particular global structure in order to model the
smooth topology of open and closed string worldsheets. We show that the
category of open-closed TQFTs is equivalent to the category of knowledgeable
Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a
symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an
algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions.
This result is achieved by providing a generators and relations description of
the category of open-closed cobordisms. In order to prove the sufficiency of
our relations, we provide a normal form for such cobordisms which is
characterized by topological invariants. Starting from an arbitrary such
cobordism, we construct a sequence of moves (generalized handle slides and
handle cancellations) which transforms the given cobordism into the normal
form. Using the generators and relations description of the category of
open-closed cobordisms, we show that it is equivalent to the symmetric monoidal
category freely generated by a knowledgeable Frobenius algebra. Our formalism
is then generalized to the context of open-closed cobordisms with labeled free
boundary components, i.e. to open-closed string worldsheets with D-brane labels
at their free boundaries.
