The effect of chemotaxis on migration of adhesive and proliferative cells on a substrate is analyzed by employing two approaches: by solving a stochastic discrete lattice model for cell dynamics and by deriving and solving a continuum macroscopic equation for cell density. The phenomenon of front propagation is investigated in the framework of the two approaches both for positive and negative chemotaxis. A good agreement between the results of the lattice model and of the continuum model is observed both for front velocities and front profiles. The theoretical model is also able to match recent experimental observations on glioma cell migration.