  In this paper we show that not all affine rational complex surfaces can be
parametrized birationally and surjectively. For this purpose, we prove that, if
S is an affine complex surface whose projective closure is smooth, a necessary
condition for S to admit a birational surjective parametrization from an open
subset of the affine complex plane is that the infinity curve of S must contain
at least one rational component. As a consequence of this result we provide
examples of affine rational surfaces that do not admit birational surjective
parametrizations.
