  A two--parameter family of asymmetric exclusion processes for particles on a
one-dimensional lattice is defined. The two parameters of the model control the
driving force and an effect which we call pushing, due to the fact that
particles can push each other in this model. We show that this model is exactly
solvable 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 in
determining various steady state and dynamical characteristics of the system.
