This book deals with the mathematical analysis and the numerical approximation of eddy current problems in the time-harmonic case. All the most used formulations are taken into account, placing the problem in a rigorous functional framework. Nodal or edge finite elements are used for approximation. A detailed analysis of each formulation is presented, focusing on the well-posedness and on the efficiency of numerical schemes. A particular attention is devoted to the topology of the physical domain. Some specific applications to real-life problems are described. The reader will also find a complete presentation of some recent and new results on the subject.