  We introduce and study a universal model of random geometry in two
dimensions. To this end, we start from a discrete graph drawn on the sphere,
which is chosen uniformly at random in a certain class of graphs with a given
size $n$, for instance the class of all triangulations of the sphere with $n$
faces. We equip the vertex set of the graph with the usual graph distance
rescaled by the factor $n^{-1/4}$. We then prove that the resulting random
metric space converges in distribution as $n\to\infty$, in the Gromov-Hausdorff
sense, toward a limiting random compact metric space called the Brownian map,
which is universal in the sense that it does not depend on the class of graphs
chosen initially. The Brownian map is homeomorphic to the sphere, but its
Hausdorff dimension is equal to $4$. We obtain detailed information about the
structure of geodesics in the Brownian map. We also present the infinite-volume
variant of the Brownian map called the Brownian plane, which arises as the
scaling limit of the uniform infinite planar quadrangulation. Finally, we
discuss certain open problems. This study is motivated in part by the use of
random geometry in the physical theory of two-dimensional quantum gravity.
