In recent years an increasing interest in functional or smart materials such as ferroelectric polymers and ceramics has been shown. Regarding the technical implementation of smart systems a broad variety of physically-based phenomena and materials are available, where some of the most important coupling effects are the shape memory effect, magnetostriction, electrostriction, and piezoelectricity. Typical fields of application are adaptive or controlled systems such as actuators and sensors, micro-electro-mechanical systems (MEMS), fuel injectors for common rail diesel engines, ferroelectric random access memories, and artificial muscles used in robotics. A highly interesting class of these materials are piezoceramics, coming up with short response times, high precision positioning, relatively low power requirements, and high generative forces, providing an excellent opportunity for mass production. Typical examples of such materials are barium titanate and lead zirconate titanate crystals and polycrystals, which exhibit linear and nonlinear coupling phenomena as well as hysteresis under high cyclic loading. At the microscale level, these materials are composed of several homogeneously polarized regions, called ferroelectric domains, whose evolution in time is driven by external electric fields and stresses applied to a sample of the material. Ferroelectric domains are regions of parallel and hence aligned polarization. Electric poling can be achieved by the application of a sufficiently strong electric field, inducing the reorientation and alignment of spontaneous polarization. As a consequence, piezoceramics exhibit a macroscopic remanent polarization. On the other hand, there are electroactive polymers (EAPs) responding by a (possibly large) deformation to an applied electrical stimulus, an effect discovered by the physicist Wilhem Röntgen in 1880 in an experiment on a rubber strip subjected to an electric field. They are divided into two main groups: electronic and ionic materials. The description of these effects through models of continuum physics is a subject of extensive research. Physically predictive material modeling can be performed on different length- and time scales. The classical setting of continuum mechanics develops phenomenological material models "smeared" over some continuously distributed material, where the material parameters are determined from experimental data. Nowadays developed multiscale techniques focus predominantly on the efficient bridging of neighboring length- and time scales, e.g. the incorporation of the microscopic polarization in order to predict macroscopic hysteresis phenomena. With a continuous increase in computational power and the development of efficient numerical solvers, real multiscale simulations seem to be a reachable goal. Computational homogenization schemes determine, in contrast to initially developed Voigt and Reuss bounds, the effective properties numerically. No constitutive model is explicitly assumed at the macroscale, and the material response at each point is determined by performing a separate numerical analysis at the micro-level. The macroscopic material behavior in this two-scale scenario is then determined by separate FE computations at the microscale. Main ingredients of such a framework are, on the one hand, the solution of a microscopic material model describing mechanical behavior at the representative volume element and, on the other hand, a homogenization rule determining the macroscopic stress tensor by its microscopic counterpart. Goal of these computational homogenization techniques is the modeling of the overall response based on well-defined microstructural information. Concerning the scale transition for functional materials, it is necessary to extend the homogenization principles to coupled problems, incorporating besides the mechanical displacement further primary variables such as the electric potential and the electric polarization. The key aspect of every homogenization scheme is the determination of macroscopic quantities in terms of their microscopic counterpart, driven by appropriate constraints or boundary conditions on the representative volume element. The micro-to-macro transition can be described in a canonical manner by variational principles of homogenization, determining macroscopic potentials in terms of their microscopic counterparts.