Fluid flow over a moving semi-infinite flat surface with temperature dependent viscosity has been investigated in this study. A steady two-dimensional laminar boundary layer flow of incompressible, Newtonian fluid past a flat plate has been considered. The surface is considered to move with uniform velocity Uw in the same direction to the fluid with velocity U in the free stream region. The surface is kept at constant uniform temperature Tw higher than the fluid temperature T at the free stream region such that the temperature difference causes variation of viscosity within the boundary layer region. This flow is considered to be caused by constant pressure gradient and the movement of the surface. The study aims at determining the effect of varying various flow parameters on velocity and temperature profiles.These parameters are pressure gradient,Reynolds number, Eckert number, variable viscosity parameter , Prandtl number and surface velocity. The partial differential equations governing the flow have been non-dimensionalised and then solved using finite difference numerical method. The results obtained have been presented graphically and in form of tables. It has been observed that an increase in pressure gradient, Reynolds number , variable viscosity parameter and surface velocity increase both primary and secondary velocity profiles whereas it decreases temperature profiles. Increase in Eckert number has been observed to increase temperature profiles whereas increase in Prandtl number was noted to decrease temperature profiles. These results are useful in paper and polymer production where hot filaments are cooled as they pass through a moving fluid.