This chapter introduces the rudiments of distribution theory and the Fourier transform, with a brief exploration of oscillatory integrals. The treatment is meant as a recap rather than an exhaustive exposition. We highlight the shift in perspective introduced by distributions, developed by Laurent Schwartz, where functions are identified with linear functionals. This approach extends classical notions, such as derivatives, through duality principles, significantly broadening their applicability. We also present key examples, such as the Dirac delta, which illustrates the limitations of classical analysis. This foundational material bridges the gap between classical analysis and the advanced tools used in subsequent chapters. It equips readers with essential concepts for understanding the broader applications of distributions in mathematical analysis and partial differential equations.