: Two jewels of classical physics are James Clerk Maxwell’s field equations (1861–1862) and the Lorentz force law (Henrick Lorentz, 1892) for electrodynamics. Although this field theory is not exact in some extreme situations, it is bedrock physics that has been used successfully to solve many problems in applied science. A brief introduction to electromagnetism from a mathematical point of view is presented, and the electromagnetic boundary value problem is proved to have unique solutions. Time-harmonic fields are introduced. The limitations of Maxwell’s theory are discussed.