  We define A-infinity-bimodules similarly to Tradler and show that this notion
is equivalent to an A-infinity-functor with two arguments which takes values in
the differential graded category of complexes of k-modules, where k is a ground
commutative ring. Serre A-infinity-functors are defined via
A-infinity-bimodules likewise Kontsevich and Soibelman. We prove that a unital
closed under shifts A-infinity-category A over a field k admits a Serre
A-infinity-functor if and only if its homotopy category H^0(A) admits a Serre
k-linear functor. The proof uses categories enriched in K, the homotopy
category of complexes of k-modules, and Serre K-functors. Also we use a new
A-infinity-version of the Yoneda Lemma generalizing the previously obtained
result.
