  The search for a superconductor with non-s-wave pairing is important not only
for understanding unconventional mechanisms of superconductivity but also for
finding new types of quasiparticles such as Majorana bound states. Materials
with both topological band structure and superconductivity are promising
candidates as $p+ip$ superconducting states can be generated through pairing
the spin-polarized topological surface states. In this work, the electronic and
phonon properties of the superconductor molybdenum carbide (MoC) are studied
with first-principles methods. Our calculations show that nontrivial band
topology and superconductivity coexist in both structural phases of MoC,
namely, the cubic $\alpha$ and hexagonal $\gamma$ phases. The $\alpha$ phase is
a strong topological insulator and the $\gamma$ phase is a topological nodal
line semimetal with drumhead surface states. In addition, hole doping can
stabilize the crystal structure of the $\alpha$ phase and elevate the
transition temperature in the $\gamma$ phase. Therefore, MoC in different
structural forms can be a practical material platform for studying topological
superconductivity and elusive Majorana fermions.
