  A mathematical framework of cohomological field theories (CohFTs) is
formulated in the language of bigraded manifolds. Algebraic properties of
operators in CohFTs are studied. Methods of constructing CohFTs, with or
without gauge symmetries, are discussed. In particular, a generalization of the
Mathai-Quillen formalism is given. Examples such as topological quantum
mechanics, topological sigma model, topological M-theory, and topological
Yang-Mills theory can be obtained uniformly using this new formalism.
