  In a Dirac semimetal, the conduction and valence bands contact only at
discrete (Dirac) points in the Brillouin zone (BZ) and disperse linearly in all
directions around these critical points. Including spin, the low energy
effective theory around each critical point is a four band Dirac Hamiltonian.
In two dimensions (2D), this situation is realized in graphene without
spin-orbit coupling. 3D Dirac points are predicted to exist at the phase
transition between a topological and a normal insulator in the presence of
inversion symmetry. Here we show that 3D Dirac points can also be protected by
crystallographic symmetries in particular space-groups and enumerate the
criteria necessary to identify these groups. This reveals the possibility of 3D
analogs to graphene. We provide a systematic approach for identifying such
materials and present ab initio calculations of metastable \beta-cristobalite
BiO_2 which exhibits Dirac points at the three symmetry related X points of the
BZ.
