  The binary Re$_{1-x}$Mo$_x$ alloys, known to cover the full range of solid
solutions, were successfully synthesized and their crystal structures and
physical properties investigated via powder x-ray diffraction, electrical
resistivity, magnetic susceptibility, and heat capacity. By varying the Re/Mo
ratio we explore the full Re$_{1-x}$Mo$_x$ binary phase diagram, in all its
four different solid phases: hcp-Mg ($P6_3/mmc$), $\alpha$-Mn
($I\overline{4}3m$), $\beta$-CrFe ($P4_2/mnm$), and bcc-W ($Im\overline{3}m$),
of which the second is non-centrosymmetric with the rest being centrosymmetric.
All Re$_{1-x}$Mo$_x$ alloys are superconductors, whose critical temperatures
exhibit a peculiar phase diagram, characterized by three different
superconducting regions. In most alloys the $T_c$ is almost an order of
magnitude higher than in pure Re and Mo. Low-temperature electronic
specific-heat data evidence a fully-gapped superconducting state, whose
enhanced gap magnitude and specific-heat discontinuity suggest a moderately
strong electron-phonon coupling across the series. Considering that several
$\alpha$-Mn-type Re$T$ alloys ($T$ = transition metal) show time-reversal
symmetry breaking (TRSB) in the superconducting state, while TRS is preserved
in the isostructural Mg$_{10}$Ir$_{19}$B$_{16}$ or Nb$_{0.5}$Os$_{0.5}$, the
Re$_{1-x}$Mo$_x$ alloys represent another suitable system for studying the
interplay of space-inversion, gauge, and time-reversal symmetries in future
experiments expected to probe TRSB in the Re$T$ family.
