  We define the tensor product of filtered $A_\infty$-algebras. establish some
of its properties and give a partial description of the space of bounding
cochains in the tensor product. Furthermore we show that in the case of
classical $A_\infty$-algebras our definition recovers the one given by Markl
and Shnider. We also give a criterion that implies that a given
$A_\infty$-algebra is quasi-isomorphic to the tensor product of two
subalgebras. This will be used in a sequel to prove a K\"unneth Theorem for the
Fukaya algebra of a product of Lagrangian submanifolds.
