We present a model for crawling cell motion in 2D and discuss steady state solutions (corresponding to a resting cell) and traveling wave solutions (steadily moving cells). Traveling waves will be shown to emerge via bifurcation from the steady state solution branch. To show the bifurcation mechanism via mapping degree theory we will start with a 1D version of the model as an illustrative example. We will then formulate the traveling wave problem in 2D and show the existence of a continuum of steady state solutions parametrized by an appropriately chosen parameter. Using this parameter as bifurcation parameter we will then discuss the bifurcation conditions in 2D. In addition to the analytical investigation some numerical simulations will be presented which allow quantitatively graphing the relevant bifurcation diagrams for different physical properties of the cell encoded in the model parameters.