The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as $λ^{(p)}(G)=\max \{x^TAx : \|x\|_p=1 \}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix $L$, and define the $p$-spectral radius of the Laplacian as $μ^{(p)}(G)=\max \{x^TLx : \|x\|_p=1 \}$. We show that $μ^{(p)}(G)$ relates to invariants such as maximum degree and size of a maximum cut. We also show properties of $μ^{(p)}(G)$ as a function of $p$, and a upper bound on $\max_{G \colon |V(G)|=n} μ^{(p)}(G)$ in terms of $n=|V|$ for $p\ge 2$, which is attained if $n$ is even.