This paper explores the application of noncommutative geometry to the compactification of Matrix theory on tori. The authors study the compactification of Matrix theory on both commutative and noncommutative tori, and argue that these backgrounds correspond to tori with constant background three-form tensor fields in supergravity. They introduce the IKKT and BFSS Matrix models, and discuss their relation to superstring theory. The paper includes an introduction to the IKKT model and its relation to Green-Schwarz superstring theory. The authors show that the defining relations of toroidal compactification in the framework of [1] are precisely the definition of a connection on the noncommutative torus. They discuss the classification of these backgrounds and argue that they correspond to constant curvature connections on the noncommutative torus. The paper also discusses the physical interpretation of these compactifications in the context of the BFSS model. The authors show that deforming the commutative torus to the noncommutative torus corresponds to turning on a constant background three-form potential. They also show that the additional SL(2,ℤ) duality symmetry predicted by the noncommutative geometry approach is present and corresponds to T-duality on a two-torus including the compact null dimension. The paper concludes with a discussion of the moduli space of constant curvature connections and its relation to the moduli space of supersymmetric vacua in eleven-dimensional supergravity.This paper explores the application of noncommutative geometry to the compactification of Matrix theory on tori. The authors study the compactification of Matrix theory on both commutative and noncommutative tori, and argue that these backgrounds correspond to tori with constant background three-form tensor fields in supergravity. They introduce the IKKT and BFSS Matrix models, and discuss their relation to superstring theory. The paper includes an introduction to the IKKT model and its relation to Green-Schwarz superstring theory. The authors show that the defining relations of toroidal compactification in the framework of [1] are precisely the definition of a connection on the noncommutative torus. They discuss the classification of these backgrounds and argue that they correspond to constant curvature connections on the noncommutative torus. The paper also discusses the physical interpretation of these compactifications in the context of the BFSS model. The authors show that deforming the commutative torus to the noncommutative torus corresponds to turning on a constant background three-form potential. They also show that the additional SL(2,ℤ) duality symmetry predicted by the noncommutative geometry approach is present and corresponds to T-duality on a two-torus including the compact null dimension. The paper concludes with a discussion of the moduli space of constant curvature connections and its relation to the moduli space of supersymmetric vacua in eleven-dimensional supergravity.