The aim of this mathematical monography is to give a treatment of the theory of functions of one quaternionic variable, starting with the algebraic construction of the division ring of the quaternions and ending with a derivation of several extensions of famous theorems of complex analysis to the quaternions (such as the Cauchy-Riemann equations, Cauchy-Goursat's theorem, Morera's theorem, Taylor and Laurent series expansions exc.). We will also focus on the geometric properties of quaternions, such as their ability to represent 3-dimensional rotations and the ways in which they're employed for spherical interpolation of rotations. The text, moreover, contains numerous examples and exercises, as they're a crucial part of the learning process. To make this text accessible to as much people as possible, the latter also presents a quick recap of all the general topology, algebraic topology and differential geometry needed to understand the text.