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 (μSR) 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, ranging from pure rhenium, to ReT (T = 3d-5d early transition metals), to the dilute-Re case of ReBe₂₂. μSR 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.