  It is well-known that classical two-dimensional topological field theories
are in one-to-one correspondence with commutative Frobenius algebras. An
important extension of classical two-dimensional topological field theories is
provided by open-closed two-dimensional topological field theories.
  In this paper we extend open-closed two-dimensional topological field
theories to nonorientable surfaces. We call them Klein topological field
theories(KTFT).
  We prove that KTFTs bijectively correspond to algebras with certain
additional structures, called structure algebras.
  Semisimple structure algebras are classified.
  Starting from an arbitrary finite group, we construct a structure algebra and
prove that it is semisimple.
  We define an analog of Hurwitz numbers for real algebraic curves and prove
that they are correlators of a KTFT. The structure algebra of this KTFT is the
structure algebra of the symmetric group.
