A continuum model of cell motility in ameboid cells based on a viscoelastic description of the cytoplasm and active stress generation controlled by extracellular signals is developed and analyzed. The characteristics of locomotion depend on the specific active stress, elastic and viscous properties of the cytoplasm as well as on the strength of cell-substrate interactions. A one-dimensional version of the model is applied to describe the motion of a fibroblast. The force balance equation for the cell is solved together with reaction diffusion equations describing the dynamics of proteins essential for cell locomotion. The cell deformation is calculated, and the deformation patterns observed experimentally are reproduced by the model. The cell velocity as a function of cell-substrate interaction is also computed for various cell characteristics such as the active stress generated, the cell elasticity and the coupling between cell-substrate interaction and the ability of the cell to contract.