The theory of modular forms is a branch of mathematics that studies complex functions with specific symmetry properties. These functions are defined on the upper half of the complex plane and exhibit behavior that is invariant under transformations from the modular group, which consists of certain linear fractional transformations. Modular forms have applications in number theory, particularly in understanding the properties of integers and prime numbers. One of the key aspects of modular forms is their connection to elliptic curves and number theory. They play a crucial role in the proof of Fermat's Last Theorem by Andrew Wiles, linking deep areas of mathematics. Modular forms also contribute to the study of automorphic forms and have implications in mathematical physics and cryptography.