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.