  We study geodesics in the random metric space called the Brownian map, which
appears as the scaling limit of large planar maps. In particular, we completely
describe geodesics starting from the distinguished point called the root, and
we characterize the set S of all points that are connected to the root by more
than one geodesic. The set S is dense in the Brownian map and homeomorphic to a
non-compact real tree. Furthermore, for every x in S, the number of distinct
geodesics from x to the root is equal to the number of connected components of
the complement of {x} in S. In particular, points of the Brownian map can be
connected to the root by at most three distinct geodesics. Our results have
applications to the behavior of geodesics in large planar maps.
