  In the recent search for unconventional- and topological superconductivity,
noncentrosymmetric superconductors (NCSCs) rank among the most promising
candidate materials. Surprisingly, some of them -- especially those containing
rhenium -- seem to exhibit also time-reversal symmetry (TRS) breaking in their
superconducting state, while TRS is preserved in many other isostructural
NCSCs. To date, a satisfactory explanation for such discrepant behavior, albeit
crucial for understanding the unconventional superconductivity of these
materials, is still missing. Here we review the most recent developments
regarding the Re-based class, where the muon-spin relaxation ($\mu$SR)
technique plays a key role due to its high sensitivity to the weak internal
fields associated with the TRS breaking phenomenon. We discuss different cases
of Re-containing superconductors, comprising both centrosymmetric- and
noncentrosymmetric crystal structures and ranging from pure rhenium, to Re$T$
($T$ = 3$d$-5$d$ early transition metals), to the dilute-Re case of
ReBe$_{22}$. $\mu$SR results suggest that the rhenium presence and its amount
are two key factors for the appearance and the extent of TRS breaking in
Re-based superconductors. Besides summarizing the existing findings, we also
put forward future research ideas regarding the exciting field of materials
showing TRS breaking.
