  Motivated by the descent equation in string theory, we give a new
interpretation for the action of the symmetry charges on the BRST cohomology in
terms of what we call {\em the Gerstenhaber bracket}. This bracket is
compatible with the graded commutative product in cohomology, and hence gives
rise to a new class of examples of what mathematicians call a {\em Gerstenhaber
algebra}. The latter structure was first discussed in the context of Hochschild
cohomology theory \cite{Gers1}. Off-shell in the (chiral) BRST complex, all the
identities of a Gerstenhaber algebra hold up to homotopy. Applying our theory
to the c=1 model, we give a precise conceptual description of the
BRST-Gerstenhaber algebra of this model. We are led to a direct connection
between the bracket structure here and the anti-bracket formalism in BV theory
\cite{W2}. We then discuss the bracket in string backgrounds with both the left
and the right movers. We suggest that the homotopy Lie algebra arising from our
Gerstenhaber bracket is closely related to the HLA recently constructed by
Witten-Zwiebach. Finally, we show that our constructions generalize to any
topological conformal field theory.
