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A two--parameter family of asymmetric exclusion processes for particles on aone-dimensional lattice is defined. The two parameters of the model control thedriving force and an effect which we call pushing, due to the fact thatparticles can push each other in this model. We show that this model is exactlysolvable via the coordinate Bethe Ansatz and show that its {\it N}-particle$S$-matrix is factorizable. We also study the interplay of the above effects indetermining various steady state and dynamical characteristics of the system.Comment: The last version that will be appeared in " Jour. Stat. Phys. (1999)

A two--parametric family of asymmetric exclusion processes and its exact solution