  The purpose of this paper is to discuss the universal algebra theory of
hom-algebras. This kind of algebra involves a linear map which twists the usual
identities. We focus on hom-associative algebras and hom-Lie algebras for which
we review the main results. We discuss the envelopment problem, operads, and
the Diamond Lemma; the usual tools have to be adapted to this new situation.
Moreover we study Hilbert series for the hom-associative operad and free
algebra, and describe them up to total degree equal 8 and 9 respectively.
