Dielectric elastomers (DEs) are an important class of materials, currently employed in the design and realization of electrically-driven, highly deformable actuators and devices, which find application in several fields of technology and engineering, including aerospace, biomedical and mechanical engineering. Subject to a voltage, a membrane of a soft dielectric elastomer coated by compliant electrodes reduces its thickness and expands its area, possibly deforming in-plane well beyond 100%: this principle is exploited to conceive transducers for a broad range of applications, including soft robots, adaptive optics, Braille displays and energy harvesters. Soft dielectrics undergo finite strains, and their modelling requires a formulation based on the Mechanics of Solids at large deformations. A major problem that limits the widespread diffusion of such devices in everyday technology is the high voltage required to activate large strains, because of the low dielectric permittivity of typical materials (acrylic elastomers or silicones), in the order of few unities, which governs the electromechanical coupling. Composite materials (reinforcing a soft matrix with stiff and high-permittivity particles) provide a way to overcome these limitations, as suggested by some experiments. In addition, composites can display failure modes and instabilities not displayed by homogeneous specimens that must be thoroughly investigated. Commonly, instability phenomena are seen as a serious drawback, that should be predicted and avoided. However, in some cases they can be used to activate snap-through actuation, as in the case of buckling-like or highly-deformable balloon-like actuators. Soft dielectric elastomers display electrostrictive properties (permittivity depending on the deformation) and we show how to take into account such a phenomenon within the theory of electroelasticity. Original results regard the investigation of diffuse modes (buckling like instabilities etc.), surface mode instabilities and localized modes. New (analytical) solutions for band-localization instability are provided and then it has been investigated how such instabilities are related to electrostriction. Regarding DE composites, the goal is to evaluate in detail the behaviour of two-phase rank-1 laminates in terms of different types of actuation, geometric and mechanical properties of phases, applied boundary conditions, and instabilities phenomena, in order to establish precise ranges in which the performance enhancement is effective with respect to the homogeneous counterpart.