Let $G_{1}$, $G_{2}$ be two connected graphs. Denote the corona and the edge corona of $G_{1}$, $G_{2}$ by $G_1\circ G_2$ and $G=G_{1}\diamond G_{2}$, respectively. In this paper, we compute the Laplacian spectrum of the corona $G\circ H$ of two arbitrary graphs $G$ and $H$ and the edge corona of a connected regular graph $G_{1}$ and an arbitrary graph $G_{2}$.