On the integer lattice we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in $d\ge 2$. Namely, it is shown that the scaling limit in $d=2,\,3$ is a Hölder continuous random field, while in $d\ge 4$ the membrane model converges to a random distribution. As a by-product of the proof in $d=2,\,3$, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel (2009) in $d=1$.