  We study theoretically the interplay between magnetism and topology in
three-dimensional HgTe/MnTe superlattices stacked along the (001) axis. Our
results show the evolution of the magnetic topological phases with respect to
the magnetic configurations. An axion insulator phase is observed for the
antiferromagnetic order with the out-of-plane N\'eel vector direction below a
critical thickness of MnTe, which is the ground state amongst all magnetic
configurations. Defining $T$ as the time-reversal symmetry, this axion
insulator phase is protected by a magnetic two-fold rotational symmetry
$C_2{\cdot}T$. The axion insulator phase evolves into a trivial insulator as we
increase the thickness of the magnetic layers. By switching the N\'eel vector
direction into the $ab$ plane, the system realizes different antiferromagnetic
topological insulators depending on the thickness of MnTe. These phases feature
gapless surface Dirac cones shifted away from high-symmetry points on surfaces
perpendicular to the N\'eel vector direction of the magnetic layers. In the
presence of ferromagnetism, the system realizes a magnetic Weyl semimetal and a
ferromagnetic semimetal for out-of-plane and in-plane magnetization directions,
respectively. We observe large anomalous Hall conductivity in the presence of
ferromagnetism in the three-dimensional superlattice.
