This thesis constitutes a comprehensive study of the fundamental properties of an analytic function theory over quaternions, which is known as slice regularity, as introduced in Gentili et al. (Gentili et al., 2022). In 2006, an approach to analysis over quaternions was introduced by G. Gentili and D.C. Struppa. This approach gave rise to a new notion of regularity for quaternion-valued functions called slice regularity. Functions satisfying this regularity were termed slice-regularity functions. This new theory heavily relies on complex analysis and has found applications in geometry. The primary objective of this project is to present this new theory, encompassing polynomials and power series in slice regular functions with quaternionic coefficients. Various properties, such as slice regular products, extend to polynomials, and convergent power series in the form of slice regular functions are defined.Consequently, this thesis offers an overview of the latest developments in regularity theory and the fundamentals of quaternion-valued slice regularity functions. I assume the reader isfamiliar with the basics of complex analysis or has read my bachelor’s thesis (Fathian Pourkondori, 2022) in advance. Given that my master’s thesis builds upon the foundations laidin my bachelor’s thesis, I have refrained from rewriting certain chapters on quaternions and Fueter’s regularity theory. These chapters are prerequisites for understanding slice regularity, and I provide the link to the previous thesis (Fathian Pourkondori, 2022) for reference.