  A new transparent proof of the well known good compactification theorem for
the complex torus $(\Bbb C^*)^n$ is presented. This theorem provides a powerful
tool in enumerative geometry for subvarieties in the complex torus. The paper
also contains an algorithm constructing a good compactification for a
subvariety in $(\Bbb C^*)^n$ explicitly defined by a system of equations. A new
theorem on a torodoidal like compactification is stated. A transparent proof of
this generalization of the good compactification theorem which is similar to
proofs and constructions from this paper will be presented in a forthcoming
publication.
