In this talk we present recent results on deformations of canonical morphisms of varieties of general type and some applications. The applications include the description of moduli components and consequences for the geometry of Calabi Yau varieties of arbitrary dimension. To accomplish the above, we deal with the more general setting of 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 give a criterion that determines when a finite map can be deformed to a one-to-one map. We use this general result that holds in all dimensions to construct new varieties of general type with a birational canonical map. Specializing to surfaces, the results address a question that Enriques posed in 1944 for the case of an algebraic surface. Most of known families until now were complete intersections or divisors in three folds, while the varieties we construct are not of this type. Speaker(s): Bangere Purnaprajna (University of Kansas)