  We have performed an experimental study on the temperature dependence of
electrical resistivity $\rho$($T$) and heat capacity $C$($T$) of the
Mo$_{1-x}$Re$_x$ $(x = 0.20, 0.25)$ alloy superconductors in different magnetic
fields. In the presence of applied magnetic field, the electrical resistivity
of these alloys go to zero at a temperature well above the bulk superconducting
transition temperature obtained with the help of heat capacity measurements in
the same magnetic field. Our study indicates the presence of surface
superconducting state in these alloys, where the flux lines are pinned in the
surface sheath of the superconductor. The configuration of the flux-lines (2d
pancake-like) in the surface sheath is understood in the realm of the flux-spot
model. Experimental evidence in support of the surface mixed-state state or
"Kulik vortex-state" and the occurrence of a vortex-liquid to vortex-glass
transition is presented.
