Superoscillations are a phenomenon where a band-limited signal locally oscillates faster than its highest frequency component, achieved by combining low-frequency waves to mimic high-frequency oscillations. They are important because they enable super-resolution imaging and play a significant role in weak values in quantum mechanics. The aim of this chapter is to summarize some of the main facts of the mathematics of the theory of superoscillations. Superoscillations will be defined as sequences of holomorphic functions that admit integral representations with respect to complex Borel measures and converge to a plane wave in the space \(\mathcal {A}_1(\mathbb {C})\) of exponentially bounded entire functions. Additionally, it aims to explain the main results obtained in the study of the evolution of superoscillations via the Schrödinger equation.