The behaviour of the critical current density, Jc, of PbMo6Ss and SnMo6S8 monofilamentary wires with external transverse compressive stress, at, and axial tensile stress, aa, was investigated. Varying the reinforcing steel content in the wire matrix, different axial prestress states, e.g. the compressive, tensile and unstressed state were realized. The observed variation of Jc versus cra was significantly smaller compared with that of Nb3Sn wires, which is explained by the higher upper critical magnetic field, Be2, of the Chevrel phase. The elastic strain range of 0.6-0.8% (with 0.2% compressive prestrain of the filament) fulfils the requirements for future technical applications of this type of wire material. The effect of transverse applied compressive stress, cr t, on / c and B* 2 of the Chevrel phase wires was investigated in magnetic fields up to B = 22 T and T--4.2 K. Analogous to Nb3Sn wires, a strong degradation of the critical current density with at was observed. As for the axial tensile stress, the influence of a t on Jc is found to increase with the applied magnetic field. At 20 T the transverse compressive stress effects reduce Jc about twice as much as the axial tensile stress. At present externally steel reinforced PbMosSs wires can be compared with Nb3Sn wires, the effect of transverse stress appears to be considerably smaller in the Chevrel phase wire, possibly as a consequence of the higher upper critical field with respect to Nb3Sn.