  We show that, under certain natural assumptions, large random plane bipartite
maps with a boundary converge after rescaling to a one-parameter family
($\mathrm{BD}_L$, $0 < L < \infty$) of random metric spaces homeomorphic to the
closed unit disk of $\mathbb{R}^2$, the space $\mathrm{BD}_L$ being called the
Brownian disk of perimeter $L$ and unit area. 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 $L = 0$. Similar
results are obtained for maps following a Boltzmann distribution, in which the
perimeter is fixed but the area is random.
