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, $\Gamma_R(M)$, with two distinct vertices $x,y\in T(M)^*$ forming an edge if $\operatorname{Ann}(x)\cap\operatorname{Ann}(y)\neq0$. In this paper we study the case where the torsion graph $\Gamma_R(M)$ is planar.