  We study toroidal compactification of Matrix theory, using ideas and results
of non-commutative geometry. We generalize this to compactification on the
noncommutative torus, explain the classification of these backgrounds, and
argue that they correspond in supergravity to tori with constant background
three-form tensor field. The paper includes an introduction for mathematicians
to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix
theory.
