Let \(G=(V,E)\) and \(H=(W,F)\) be graphs. The direct product \(G\times H\) of \(G\) and \(H\) is defined as follows: \(V(G\times H)=V\times W\) and \(E(G\times H)=\{\{(u,x),(v,y)\}:\{u,v\}\in E, \{x,y\}\in F\}\). In this paper the following results are obtained. If \(G\) and \(H\) are distance regular graphs of diameter at least two, then \(G\times H\) or a component of \(G\times H\) is distance regular iff each of \(G\) and \(H\) is isomorphic to \(K_{n,n}\) for some \(n\). If \(G\) is a distance regular graph of diameter at least two and \(n\geq 3\), then \(G\times K_n\) is not distance regular.