The connection between the electric and magnetic fields is fundamental to our understanding of light as electromagnetic waves. The magnetic vector potential lies at the heart of this relation. The idea emerged in the early days of research in electromagnetism but was dismissed for more than half a century until the formulation of quantum electrodynamics. The magnetic vector potential is a pivotal concept with ties to many aspects of physics and mathematics. This book unravels the nature of the magnetic vector potential, highlights its connection to quantum mechanics and superconductivity, and explores the analogy with hydrodynamics.