  In this paper we first study the moduli spaces related to Calabi-Yau
manifolds. We then apply the results to the following problem. Let $C$ be a
fixed Riemann surface with fixed finite number of points on it. Given a CY
manifold with fixed topological type, we consider the set of all families of CY
manifolds of the fixed topological type over $C$ with degenerate fibres over
the fixed points up to isomorphism. This set is called Shafarevich set. The
analogue of Shafarevich conjecture for CY manifolds is for which topological
types of CY the Shafarevich set is finite. It is well-known that the analogue
of Shafarevich conjecture is closely related to the study of the moduli space
of polarized CY manifolds and the moduli space of the maps of fixed Riemann
surface to the coarse moduli space of the CY manifolds. We prove the existence
of the Teichm\"{u}ller space of CY manifolds together with a universal family
of marked CY manifolds. From this result we derive the existence of a finite
cover of the coarse moduli space which is a non-singular quasi-projective
manifold. Over this cover we construct a family of polarized CY manifolds. We
study the moduli space of maps of the fixed Riemann with fixed points on it to
the moduli space of CY manifolds constructed in the paper such that the maps
map the fixed points on the Riemann surface to the discriminant locus. If this
moduli space of maps is finite then Shafarevich conjecture holds. We relate the
analogue of Shafarevich problem to the non-vanishing of the Yukawa coupling. We
give also a counter example of the Shafarevich problem for a class of CY
manifolds.
