We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD_L, 0 < L < 8) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD_L being called the Brownian disk of perimeter L. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where L = 0.