This book deals with the mathematical analysis and the numerical approximation of time-harmonic eddy current problems. It is self-contained and suitable for mathematicians and engineers working in the field, and also accessible for beginners. Depending on the choice of the physical unknowns, these problems are formulated in different variational ways, with specific attention to the topology of the computational domain. Finite elements of nodal or edge type are used for numerical approximation, and a complete analysis of convergence is performed. A specific feature of the book is the emphasis given to saddle-point formulations in terms of the magnetic and electric fields. New results for voltage or current intensity excitation problems are also presented.