  We give a complete description of the asymptotic behavior of a
Friedmann-Robertson-Walker Universe with ``normal'' matter and a minimally
coupled scalar field. We classify the conditions under which the Universe is or
is not accelerating. In particular, we show that only two types of large time
behavior exist: an exponential regime, and a subexponential expansion with the
logarithmic derivative of the scale factor tending to zero. In the case of the
subexponetial expansion the Universe accelerates when the scalar field energy
density is dominant and the potential behaves in a specified manner, or if
matter violates the strong energy conditon $\rho + 3p >0$. When the expansion
is exponential the Universe accelerates, and the scalar field energy density is
dominant. We also find that the existence of the Big Bang and a never ending
expansion of the Universe constrain the equation of state of matter at large
and small densities, respectively.
