  Topological Dirac semimetals are a class of semimetals that host
symmetry-protected Dirac points near the Fermi level, which arise due to a band
inversion of the conduction and valence bands. In this work, we study the less
explored class of \emph{noncentrosymmetric} topological Dirac semimetals in
three dimensions. We identify the noncentrosymmetric crystallographic point
groups required to stabilize fourfold degenerate band crossings and derive
model Hamiltonians for all distinct types of band inversions allowed by
symmetry. Using these model Hamiltonians, which emphasize the physical nature
of the allowed couplings, we establish the generic electronic phase diagram
noncentrosymmetric Dirac semimetals and show that it generically includes
phases with coexistent Weyl point nodes or Weyl line nodes. In particular, for
one specific type of band inversion in sixfold symmetric systems we show that
Weyl line nodes are always present. Based on first-principles calculations, we
predict that BiPd$_2$O$_4$ is a noncentrosymmetric Dirac semimetal under 20 Gpa
pressure and hosts topological type-II Dirac points on the fourfold rotation
axis. Furthermore, we propose that the hexagonal polar alloy
LiZnSb$_{x}$Bi$_{1-x}$ realizes a Dirac semimetal with coexistent Weyl points.
Interestingly, the emergence and location of the Weyl points is highly tunable
and can be controlled by the alloy concentration $x$. More generally, our
results not only establish band-inverted noncentrosymmetric systems as a broad
and versatile class of topological semimetals, but also provide a framework for
studying the quantum nonlinear Hall effect and nonlinear optical properties in
the Dirac semimetals.
