||The dissertation is divided into three parts. Firstly, we have established the technique of the higher dimensional jets and applied the results to study value distribution of holomorphic mappings. As applications, we have also generalized the results of holomorphic curves obtained by Ochiai, Noguchi and Green-Griffiths to the higher dimensional cases. Next, we extend the general defect relation of the associated curves of a non-degenerate holomorphic curve to moving targets. Our results also improve the defect relations of Stall for the associated mappings. Finally, we introduce some new criteria of normality and its associated Julia sets J(f) and Fatou sets F(f) for continuous mappings f on locally compact connected smooth manifolds M and discuss some basic properties of the Julia set J(f). Specifically, we have shown that J(f) [not equal to] 0 if M is compact and if [divides][italic]deg(f)divides ≥ 2, which is a generalization of a most recent result by Fornaess and Sibony about complex projective spaces. We define a new dynamical degree d(f) holomorphic mapping f : [double struck C supperscript m] → [double strck C] supperscript m] and conjecture that J(f) ≠ 0 if d(f) > 1. We also present a theorem in fixed points on [double struck C supperscript m].