Automorphic forms are complex mathematical objects that generalize the concept of functions on spaces with symmetry. They arise in number theory and algebraic geometry, particularly in the study of modular forms and L-functions. These forms exhibit invariance under certain transformations, making them useful for understanding the properties of numbers and shapes. The study of automorphic forms connects various areas of mathematics, including representation theory and arithmetic geometry. They play a crucial role in the Langlands program, which seeks to relate different areas of mathematics through deep connections between number theory and geometry.