  In this paper, we generalize the usual notions of waves, fronts and
propagation speeds in a very general setting. These new notions, which cover
all classical situations, involve uniform limits, with respect to the geodesic
distance, to a family of hypersurfaces which are parametrized by time. We prove
the existence of new such waves for some time-dependent reaction-diffusion
equations, as well as general intrinsic properties, some monotonicity
properties and uniqueness results for almost planar fronts. The classification
results, which are obtained under appropriate assumptions, show the robustness
of our general definition
