  Superoscillations are band-limited functions with the peculiar characteristic
that they can oscillate with a frequency arbitrarily faster than their fastest
Fourier component. First anticipated in different contexts, such as optics or
radar physics, superoscillations have recently garnered renewed interest after
more modern studies have successfully linked their properties to a novel
quantum measurement theory, the weak value scheme. Under this framework,
superoscillations have quickly developed into a fruitful area of mathematical
study whose applications have evolved from the theoretical to the practical
world. Their mathematical understanding, though still incomplete, recognises
such oscillations will only arise in regions where the function is extremely
small, establishing an inherent limitation to their applicability. This paper
aims to provide a detailed look into the current state of research, both
theoretical and practical, on the topic of superoscillations, as well as
introducing the two-state vector formalism under which the weak value scheme
may be realised.
