  Using the Kodaira dimension and the fundamental group of X, we succeed in
classifying algebraic surfaces which are dominable by C^2 except for certain
cases in which X is an algebraic surface of Kodaira dimension zero and the case
when X is rational without any logarithmic 1-form. More specifically, in the
case when X is compact (namely projective), we need to exclude only the case
when X is birationally equivalent to a K3 surface (a simply connected compact
complex surface which admits a globally non-vanishing holomorphic 2-form) that
is neither elliptic nor Kummer.
  With the exceptions noted above, we show that for any algebraic surface of
Kodaira dimension less than 2, dominability by C^2 is equivalent to the
apparently weaker requirement of the existence of a holomorphic image of C
which is Zariski dense in the surface. With the same exceptions, we will also
show the very interesting and revealing fact that dominability by C^2 is
preserved even if a sufficiently small neighborhood of any finite set of points
is removed from the surface. In fact, we will provide a complete classification
in the more general category of (not necessarily algebraic) compact complex
surfaces before tackling the problem in the case of non-compact algebraic
surfaces.
