  We study compactifications of Matrix theory on twisted tori and
non-commutative versions of them. As a first step, we review the construction
of multidimensional twisted tori realized as nilmanifolds based on certain
nilpotent Lie algebras. Subsequently, matrix compactifications on tori are
revisited and the previously known results are supplemented with a background
of a non-commutative torus with non-constant non-commutativity and an
underlying non-associative structure on its phase space. Next we turn our
attention to 3- and 6-dimensional twisted tori and we describe consistent
backgrounds of Matrix theory on them by stating and solving the conditions
which describe the corresponding compactification. Both commutative and
non-commutative solutions are found in all cases. Finally, we comment on the
correspondence among the obtained solutions and flux compactifications of
11-dimensional supergravity, as well as on relations among themselves, such as
Seiberg-Witten maps and T-duality.
