Ru, Min2019-11-132019-11-13December 22016-12December 2https://hdl.handle.net/10657/5410In this dissertation, we first discuss some of the important results in Nevanlinna Theory and Diophantine Approximation Theory. Next, a result by the author and Min Ru \cite{LR} is presented. In chapter 3, we extend the Second Main Theorem to the case of holomorphic curves into algebraic varieties intersecting numerically equivalent ample divisors. In chapter 4, we improve Ru's defect relation (see \cite{ru15}) and the height inequality (see \cite{ru}) in the case when $X$ is a normal projective surface and $D_j$, $1 \leq j \leq q$, are big and asymptotically free divisors without irreducible common components on $X$. Lastly, the author and Gordon Heier approach the hyperbolic problem by projections from $\mathbb{P}^{n+2}$ to $\mathbb{P}^{n}$.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Navenlinna TheoryDiophantine approximationHolomorphic curves2019-11-13Thesisborn digital