  Chemical doping of topological materials may provide a possible route for
realizing topological superconductivity. However, all such cases known so far
are based on chalcogenides. Here we report the discovery of superconductivity
induced by Re doping in the topological semimetal Mo$_{5}$Si$_{3}$ with a
tetragonal structure. Partial substitution of Re for Mo in
Mo$_{5-x}$Re$_{x}$Si$_{3}$ results in an anisotropic shrinkage of the unit cell
up to the solubility limit of approximately $x$ = 2. Over a wide doping range
(0.5 $\leq$ $x$ $\leq$ 2), these silicides are found to be weakly coupled
superconductors with a fully isotropic gap. $T_{\rm c}$ increases monotonically
with $x$ from 1.67 K to 5.78 K, the latter of which is the highest among
superconductors of the same structural type. This trend in $T_{\rm c}$
correlates well with the variation of the number of valence electrons, and is
mainly ascribed to the enhancement of electron-phonon coupling. In addition,
band structure calculations reveal that superconducting
Mo$_{5-x}$Re$_{x}$Si$_{3}$ exhibits nontrivial band topology characterized by
$Z_{2}$ invariants (1;000) or (1;111) depending on the Re doping level. Our
results suggest that transition metal silicides are a fertile ground for the
exploration of candidate topological superconductors.
