  We introduce and study the random non-compact metric space called the
Brownian plane, which is obtained as the scaling limit of the uniform infinite
planar quadrangulation. Alternatively, the Brownian plane is identified as the
Gromov-Hausdorff tangent cone in distribution of the Brownian map at its root
vertex, and it also arises as the scaling limit of uniformly distributed
(finite) planar quadrangulations with n faces when the scaling factor tends to
0 less fast than n^{-1/4}. We discuss various properties of the Brownian plane.
In particular, we prove that the Brownian plane is homeomorphic to the plane,
and we get detailed information about geodesic rays to infinity.
