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.
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