A self-dual algebra is an associative or Lie algebra A together with an A bimodule isomorphism to the dual of its opposite. This induces an isomorphism of the cohomology of A with coefficients in itself with the cohomology of A in the dual of its opposite; an algebra with such an isomorphism is quasi self-dual. For these algebras H¤(A;A) is a contravariant functor of A. They form a full subcategory of the category of the category of associative or Lie algebras, respectively. Finite dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras (which are closely connected to 1+1 dimensional topological quantum field theory). Finite poset algebras are quasi self-dual.