  We present bijections for planar maps with boundaries. In particular, we
obtain bijections for triangulations and quadrangulations of the sphere with
boundaries of prescribed lengths. For triangulations we recover the beautiful
factorized formula obtained by Krikun using a (technically involved) generating
function approach. The analogous formula for quadrangulations is new. We also
obtain a far-reaching generalization for other face-degrees. In fact, all the
known enumerative formulas for maps with boundaries are proved bijectively in
the present article (and several new formulas are obtained). Our method is to
show that maps with boundaries can be endowed with certain "canonical"
orientations, making them amenable to the master bijection approach we
developed in previous articles. As an application of our enumerative formulas,
we note that they provide an exact solution of the dimer model on rooted
triangulations and quadrangulations.
