The development of efficient numerical techniques for the simulation of Maxwell equations, coupled with the steady increase on available computational resources in recent years, has paved the way for the design of microwave and optical devices with unprecedented performance and functionalities. This book describes a compatible discretization for Maxwell equations in irregular grids based on exterior differential forms and fundamental concepts from algebraic topology. This compatible discretization framework leads to finite element methods for Maxwell equations that are stable and free from spurious solutions, both in the time- and frequency-domains and in any number of dimensions. By factorizing Maxwell equations into metric and topological parts and incorporating material properties into the metric part (represented by the so-called Hodge operators), this compatible discretization approach also leads to more robust simulations in complex media such as metamaterials. This book should be of interest to researchers and practitioners in the fields of microwave and optical engineering interested in developing robust numerical algorithms for demanding applications.