  Relativistic massless Weyl and Dirac fermions exhibit the isotropic and
linear dispersion relations to preserve the Poincar\'{e} symmetry, the most
fundamental symmetry in high energy physics. In solids, the counterparts of the
Poincar\'{e} symmetry are crystallographic symmetries, and hence, it is natural
to explore generalizations of Dirac and Weyl fermions compatible with their
crystallographic symmetries and then the new physics coming along with them.
Here, we study an important kind of generalization, namely massless Dirac
fermions with higher-order dispersion relations protected by crystallographic
symmetries in three-dimensional nonmagnetic systems. We perform a systematic
search over all 230 space groups with time-reversal symmetry and spin-orbit
coupling considered. We find that the order of dispersion cannot be higher than
three, i.e., only the quadratic and cubic Dirac points (QDPs and CDPs) are
possible. We discover previously unknown classes of higher-order Dirac points,
including the chiral QDPs with Chern numbers of $\pm 4$ and the QDPs/CDPs
without centrosymmetry. Especially the chiral QDPs feature four extensive
surface Fermi arcs and four chiral Landau bands and hence leads to observable
signatures in spectroscopic and transport experiments. We further show that
these higher-order Dirac points represent parent phases for other exotic
topological structures. Via controlled symmetry breaking, QDPs and CDPs can be
transformed into double Weyl points, triple Weyl points, charge-2 Dirac points
or Weyl loops. Using first-principles calculations, we also identify possible
material candidates, including $\alpha$-TeO$_2$ and YRu$_4$B$_4$, which realize
the predicted nodal structures.
