The aim of this diploma thesis is the visualization of some fundamental results in the context of the theory of the modular group and modular functions. For this purpose the computer algebra software Mathematica is utilized. The thesis is structured in three parts. In Chapter 1, we summarize some important basic concepts of group theory which are relevant to this work. Moreover, we introduce Möbius transformations and study their geometric mapping properties. Chapter 2 is devoted to the study of the modular group from an algebraic and geometric point of view. We introduce the canonical fundamental region which gives rise to the modular tessellation of the upper half-plane. Additionally, we present a general method for finding fundamental regions with respect to subgroups of the modular group based on the concepts of 2-dimensional hyperbolic geometry. In Chapter 3 we give some concrete examples how the developed results and methods can be exploited for the visualization of certain mathematical results. Besides the visualization of function graphs of modular functions, a particularly nice result is the connection between modular transformations and continued fraction expansions.