  In this article we study the deformation of finite maps and show how to use
this deformation theory to construct varieties with given invariants in a
projective space. Among other things, we prove a criterion that determines when
a finite map can be deformed to a one--to--one map. We use this criterion to
construct new simple canonical surfaces with different $c_1^2$ and $\chi$. Our
general results enable us to describe some new components of the moduli of
surfaces of general type. We also find infinitely many moduli spaces $\mathcal
M_{(x',0,y)}$ having one component whose general point corresponds to a
canonically embedded surface and another component whose general point
corresponds to a surface whose canonical map is a degree 2 morphism.
