Cosmology `without' constants

We review varying speed of light (VSL) theories, in a cosmological and astrophysical setting, and as phenomenological descriptions of quantum gravity. We first introduce two observational puzzles, and explain how they may related to VSL. We then catalogue the various theoretical efforts associated with VSL. We focus on recent VSL theories rendering the Planck energy invariant, and capable of explaining the unwarranted existence of ultra high energy cosmic rays. We conclude highlighting the main shortcoming of current VSL theories: structure formation; and present directions in which it may be addressed.