  We give a self contained introduction to A$_\infty$-algebras,
A$_\infty$-bimodules and maps between them. The case of A$_\infty$-bimodule-map
between $A$ and its dual space $A^{*}$, which we call $\infty$-inner-product,
will be investigated in detail. In particular, we describe the graph complex
associated to $\infty$-inner-product. In a later paper, we show how
$\infty$-inner-products can be used to model the string topology BV-algebra on
the free loop space of a Poincar\'e duality space.
