Let R be a commutative ring with identity. Let M be an R-module and T(M) * be the set of nonzero torsion elements. The set T(M) * makes up the vertices of the corresponding torsion graph, Γ R (M), with two distinct vertices x, y ∈ T(M) * forming an edge if Ann(x) ∩ Ann(y) 0. In this paper we study the case where the torsion graph Γ R (M) is planar.