  In this paper, we give new criteria for affineness of a variety defined over
$\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be
singular) of dimension $d$ ($d\geq 1$) defined over $\Bbb{C}$ is an affine
variety if and only if $Y$ contains no complete curves, $H^i(Y,
{\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big
divisor, where $X$ is a projective variety containing $Y$. We construct three
examples to show that a variety is not affine if it only satisfies two
conditions among these three conditions. We also give examples to demonstrate
the difference between the behavior of the boundary divisor $D$ and the
affineness of $Y$. If $Y$ is an affine variety, then the ring $\Gamma (Y,
{\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine
variety, we do not start from this ring. We explain why we do not need to check
the noetherian property of the ring $\Gamma (Y, {\mathcal{O}}_Y)$ directly but
use the techniques of sheaf and cohomology.
