Explore the fundamental concepts of neutrinoless double beta decay in this comprehensive physics lecture delivered at the prestigious Galileo Galilei Institute. Delve into the theoretical framework and mathematical foundations that underpin this rare nuclear process, which serves as a crucial test for lepton number violation and the nature of neutrinos. Learn about the current theoretical predictions, experimental challenges, and implications for particle physics and our understanding of the universe's fundamental properties. The lecture provides an in-depth examination of the mathematical formalism, decay mechanisms, and nuclear matrix elements essential for understanding this phenomenon, making it valuable for graduate students and researchers in particle physics and nuclear theory.