  We study cohomology theories of strongly homotopy algebras, namely $A_\infty,
C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of
Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising
previous work by Loday and Gerstenhaber-Schack. These results are then used to
show that a $C_\infty$-algebra with an invariant inner product on its
cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an
$\infty$-generalisation of a commutative Frobenius algebra introduced by
Kontsevich). As another application, we show that the `string topology'
operations (the loop product, the loop bracket and the string bracket) are
homotopy invariant and can be defined on the homology or equivariant homology
of an arbitrary Poincare duality space.
