  We prove a version of Topogonov's triangle comparison theorem with surfaces
of revolution as model spaces. Given a model surface and a Riemannian manifold
with a fixed base point, we give necessary and sufficient conditions under
which every geodesic triangle in the manifold with a vertex at the base point
has a corresponding Alexandrov triangle in the model. Under these conditions we
also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type
Sphere Theorem.
