Summary: We say that a regular graph $G$ of order $n$ and degree $r\geq 1$ (which is not the complete graph) is strongly regular if there exist non-negative integers $τ$ and $θ$ such that $|S_i \cap S_j| = τ$ for any two adjacent vertices $i$ and $j$, and $|S_i \cap S_j| = θ$ for any two distinct non-adjacent vertices $i$ and $j$, where $S_k$ denotes the neighborhood of the vertex $k$. We say that a graph $G$ of order $n$ is walk regular if and only if its vertex deleted subgraphs $G_i = G \smallsetminus i$ are cospectral for $i = 1,2,\dots ,n$. We here establish necessary and sufficient conditions under which a walk regular graph $G$ which is cospectral to its complement $\overline G$ is strongly regular.