This thesis provides a logical path to develop the mathematical structures of real scalar quantum fields. It also identifies the emergence of divergences and motivates the methods to tame them (normal ordering, renormalization). We begin with classical fields and the quantum‐mechanical treatment of continuous media. These topics should be familiar for a BSc in physics. Then we will cover oscillator quantization, path integral formalism, and generating functionals. We will discuss how the addition of a self‐interaction term generates ultraviolet divergences; to deal with these, we will develop regularization and renormalization techniques. We will then extend them to a finite temperature description. As a case study, we will analyze the Sine–Gordon model, a periodic self‐interacting scalar potential.