  We study the cosmological evolution of scalar fields with arbitrary
potentials in the presence of a barotropic fluid (matter or radiation) without
making any assumption on which term dominates. We determine what kind of
potentials V(phi) permits a quintessence interpretation of the scalar field phi
and to obtain interesting cosmological results. We show that all model
dependence is given in terms of lambda= - V'/V only and we study all possible
asymptotic limits: lambda approaching zero, a finite constant or infinity. We
determine the equation of state dynamically for each case. For the first class
of potentials, the scalar field quickly dominates the universe behaviour, with
an inflationary equation of state allowing for a quintessence interpretation.
The second case gives the extensively studied exponential potential. While in
the last case, when lambda approaches infinity, if it does not oscillate then
the energy density redshifts faster than the barotropic fluid but if lambda
oscillates then the energy density redshift depends on the specific potential.
