  We study the statistical properties of geodesics, i.e. paths of minimal
length, in large random planar quadrangulations. We extend Schaeffer's
well-labeled tree bijection to the case of quadrangulations with a marked
geodesic, leading to the notion of "spine trees", amenable to a direct
enumeration. We obtain the generating functions for quadrangulations with a
marked geodesic of fixed length, as well as with a set of "confluent
geodesics", i.e. a collection of non-intersecting minimal paths connecting two
given points. In the limit of quadrangulations with a large area n, we find in
particular an average number 3*2^i of geodesics between two fixed points at
distance i>>1 from each other. We show that, for generic endpoints, two
confluent geodesics remain close to each other and have an extensive number of
contacts. This property fails for a few "exceptional" endpoints which can be
linked by truly distinct geodesics. Results are presented both in the case of
finite length i and in the scaling limit i ~ n^(1/4). In particular, we give
the scaling distribution of the exceptional points.
