A major theme in modern condensed matter physics is how topology influences the quantum phases of matter. Particularly significant progress in this field has happened within the last decade, due to the discovery of topologically nontrivial states of matter, that have a gap in their energy spectrum, namely Topological Insulators and Topological Superconductors. In this talk I will describe recent work, extending these notions to new gapless solid state phases which realize massless Dirac and Weyl fermions. These provide concrete experimental realizations of phenomena such as the famous chiral anomaly in particle physics. I will show how even simple textbook concepts, like the Fermi surface of metals, need to be revisited in the context of this topological revolution.