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In this paper, we generalize the usual notions of waves, fronts andpropagation speeds in a very general setting. These new notions, which coverall classical situations, involve uniform limits, with respect to the geodesicdistance, to a family of hypersurfaces which are parametrized by time. We provethe existence of new such waves for some time-dependent reaction-diffusionequations, as well as general intrinsic properties, some monotonicityproperties and uniqueness results for almost planar fronts. The classificationresults, which are obtained under appropriate assumptions, show the robustnessof our general definition

Generalized transition waves and their properties