The p-spectral radius of a graph G=(V,E) with adjacency matrix A is defined as ?(p)(G) = max||x||p=1 xT Ax. This parameter shows connections with graph invariants, and has been used to generalize some extremal problems. In this work, we define the p-spectral radius of the Laplacian matrix L as ?(p)(G) = max||x||p=1 xT Lx. We show that ?(p)(G) relates to invariants such as maximum degree and size of a maximum cut. We also show properties of ?(p)(G) as a function of p, and a upper bound on maxG: |V(G)|=n ?(p)(G) in terms of n = |V| for p > 2, which is attained if n is even.