The paper introduces the title subject with a description of minimal surfaces and states the classical problem of Plateau. Minimal surfaces in R³ (primarily of the type of the disc) are treated by including a brief historical account, leading into various representations of minimal surfaces. Major theorems regarding regularity of the surface and descriptions of its characteristic behavior in the neighborhood of an interior branch point are included. Interior branch points are categorized into three primary types and examples of each are discussed. Equations are given for determining the branch lines of a minimal surface with a true branch point. The subject of boundary regularity is considered and includes a detailed exposition of the proof of J. C. C. Nitsche as outlined in Inventions Mathematicae, Vol. 8 (1969), 330-333. This important proof, which establishes an asymptotic expansion for the component functions in the neighborhood of a boundary branch point, proves that the order of the branch point is even and at least two. An example of such a boundary branch point is presented. That inequality, due to Susaki, Nitsche and Lewy, which may be utilized to determine an a priori bound to the number of branch points of a minimal surface, is presented. Using this, in conjunction with Nitsche's theorem involving the order of boundary branch points, it is shown that if the total curvature of an analytic Jordan curve satisfies the inequality S R(s)ds < 4π then any minimal surface spanning must be free of boundary branch points as well as interior branch points. The expanded bibliography covers those references cited in the text. Additional references are included which are meant to provide necessary background material for a full understanding of the subject. Certain other references are given which describe work in related areas, such as; generalization of minimal surfaces to manifolds in Rⁿ and surfaces of constant (rather than zero) mean curvature.