  We perform a systematic \textit{ab initio} density functional study of the
superconductivity, electronic and phononic band structures, electron-phonon
coupling and elastic constants of all four possible structures of niobium
nitride $\beta$-Nb$_2$N as well as Nb-rich $\gamma$-Nb$_4$N$_3$ and N-rich
$\beta^\prime$-Nb$_4$N$_5$. First of all, we find that all four structures of
$\beta$-Nb$_2$N are superconductors with superconducting transition
temperatures ($T_c$) ranging from 0.6 K to 6.1 K, depending on the structure.
This explains why previous experiments reported contradicting $T_c$ values for
$\beta$-Nb$_2$N. Furthermore, both $\gamma$-Nb$_4$N$_3$ and
$\beta^\prime$-Nb$_4$N$_5$ are predicted to be superconductors with rather high
$T_c$ of 8.5 K and 15.3 K, respectively. Second, the calculated elastic
constants and phonon dispersion relations show that all the considered niobium
nitride structures are mechanically and dynamically stable. Moreover, the
calculated elastic moduli demonstrate that all the niobium nitrides are hard
materials with bulk moduli and hardness being comparable to or larger than the
well-known hard sapphire. Third, the calculated band structures reveal that the
nitrides possess both type I and type II Dirac nodal points and are thus
topological metals. Finally, the calculated electron-phonon coupling strength,
superconductivity and mechanical property of the niobium nitrides are discussed
in terms of their underlying electronic structures and also Debye temperatures.
The present \textit{ab initio} study thus indicates that $\beta$-Nb$_2$N,
$\gamma$-Nb$_4$N$_3$ and $\beta^\prime$-Nb$_4$N$_5$ are hard superconductors
with nontrivial band topology and are promising materials for exploring exotic
phenomena due to the interplay of hardness, superconductivity and nontrivial
band topology.
