  We report a study of the noncentrosymmetric TaReSi superconductor by means of
muon-spin rotation and relaxation ($\mu$SR) technique, complemented by
electronic band-structure calculations. Its superconductivity, with $T_c$ = 5.5
K and upper critical field $\mu_0H_\mathrm{c2}(0)$ $\sim$ 3.4 T, was
characterized via electrical-resistivity- and magnetic-susceptibility
measurements. The temperature-dependent superfluid density, obtained from
transverse-field $\mu$SR, suggests a fully-gapped superconducting state in
TaReSi, with an energy gap $\Delta_0$ = 0.79 meV and a magnetic penetration
depth $\lambda_0$ = 562 nm. The absence of a spontaneous magnetization below
$T_c$, as confirmed by zero-field $\mu$SR, indicates a preserved time-reversal
symmetry in the superconducting state. The density of states near the Fermi
level is dominated by the Ta- and Re-5$d$ orbitals, which account for the
relatively large band splitting due to the antisymmetric spin-orbit coupling.
In its normal state, TaReSi behaves as a three-dimensional Kramers nodal-line
semimetal, characterized by an hourglass-shaped dispersion protected by glide
reflection. By combining non\-triv\-i\-al electronic bands with intrinsic
superconductivity, TaReSi is a promising material for investigating the
topological aspects of noncentrosymmetric superconductors.
