The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let Pk be the path with k vertices, the square Pk2 of Pk is obtained by joining the pairs of vertices with distance one or two in Pk. The powerful theorem of Erdős, Stone and Simonovits determines the asymptotic behavior of ex(n,Pk2). In the present paper, we determine the exact value of ex(n,P52) and ex(n,P62) and pose a conjecture for the exact value of ex(n,Pk2).