  We present here "the" cartesian closed theory for real analytic mappings. It
is based on the concept of real analytic curves in locally convex vector
spaces. A mapping is real analytic, if it maps smooth curves to smooth curves
and real analytic curves to real analytic curves. Under mild completeness
conditions the second requirement can be replaced by: real analytic along
affine lines. Enclosed and necessary is a careful study of locally convex
topologies on spaces of real analytic mappings. As an application we also
present the theory of manifolds of real analytic mappings: the group of real
analytic diffeomorphisms of a compact real analytic manifold is a real analytic
Lie group.
