  The BRST formalism has played a fundamental role in the construction of
bosonic closed string backgrounds, ie. the stringy analogs of classical
solutions to the field equations of general relativity. The concept of a string
background has been extended to the notion of $W$-strings, where the BRST
symmetry is still largely conjectural. More recently, the BRST formalism has
entered the construction of two dimensional topological conformal quantum field
theories, such as those that arise from Calabi-Yau varieties. In this lecture,
we focus on common features of the BRST cohomology algebras of string
backgrounds and topological field theories. In this context, we present some
new evidence for a remarkable relationship that transports us from bosonic and
$W$-string backgrounds to the B-model topological conformal field theories
associated to certain noncompact Calabi-Yau varieties. This paper will appear
in the proceedings of the {\it Symposium on BRS Symmetry} held at RIMS,
September 18-22, 1995.
