  An n-dimensional complex manifold M is said to be (holomorphically) dominable
by $\CC^n$ if there is a map $F:\CC^n \ra M$ which is holomorphic such that the
Jacobian determinant $\det(DF)$ is not identically zero. Such a map F is called
a dominating map. In this paper, we attempt to classify algebraic surfaces X
which are dominable by $\CC^2$ using a combination of techniques from algebraic
topology, complex geometry and analysis. One of the key tools in the study of
algebraic surfaces is the notion of Kodaira dimension (defined in section 2).
By Kodaira's pioneering work and its extensions, an algebraic surface which is
dominable by $\CC^2$ must have Kodaira dimension less than two. Using the
Kodaira dimension and the fundamental group of X, we succeed in classifying
algebraic surfaces which are dominable by $\CC^2$ 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 (see sections 3 and 4 for the definition of these
types of surfaces).
