We will review remarkable recent progress in both experimental and theoretical research on flat bands materials. When a system exhibits flat bands despite having sizable hopping integrals/electron itineracy, the flat band is likely to be topological. In stoichiometric materials, one now has flat band catalogues of materials exhibiting such electronic states at the Fermi level. I will review the interacting and topological physics of such bands, first with simple one-body models and then adding interactions to obtain Kondo physics, magnets, charge density waves, and superconductors. In engineered moire materials, where the Fermi level can be gate-tuned, a new degree of precise controllability allows for the development of flat bands exhibiting Chern insulators, Superconductivity, and - most recently - Fractional Chern insulators. We will show that the physics of most of these bands holds a close resemblance to heavy Fermion physics