  We report a comprehensive study of the centrosymmetric Re$_3$B and
noncentrosymmetric Re$_7$B$_3$ superconductors. At a macroscopic level, their
bulk superconductivity (SC), with $T_c$ = 5.1 K (Re$_3$B) and 3.3 K
(Re$_7$B$_3$), was characterized via electrical-resistivity, magnetization, and
heat-capacity measurements, while their microscopic superconducting properties
were investigated by means of muon-spin rotation/relaxation ($\mu$SR). In both
Re$_3$B and Re$_7$B$_3$ the low-$T$ zero-field electronic specific heat and the
superfluid density (determined via tranverse-field $\mu$SR) suggest a nodeless
SC. Both compounds exhibit some features of multigap SC, as evidenced by
temperature-dependent upper critical fields $H_\mathrm{c2}(T)$, as well as by
electronic band-structure calculations. The absence of spontaneous magnetic
fields below the onset of SC, as determined from zero-field $\mu$SR
measurements, indicates a preserved time-reversal symmetry in the
superconducting state of both Re$_3$B and Re$_7$B$_3$. Our results suggest that
a lack of inversion symmetry and the accompanying antisymmetric spin-orbit
coupling effects are not essential for the occurrence of multigap SC in these
rhenium-boron compounds.
