Searches for a permanent Electric Dipole Moment (EDM) of a fundamental particle provide a wide window for the discovery of potential New Physics. Within todays Standard Model in particle physics the well established violation of CP symmetry gives rise to EDMs which are several orders of magnitude below the present experimentally established upper bounds. On the other hand, EDMs appear quite naturally within many modern speculative theories, which have been suggested to improve the known shortcomings of the present Standard Model, e.g., the lack of giving reasons for certain established facts such as the mass hierarchy of the fundamental fermions or the number of three particle generations. They could be almost as large as the present experimental bounds. The speculative models provide for EDMs of different fundamental particles in specific ways. As there is no convincing indication, yet, which of the various extensions to the present standard theory may be more successful, a larger number of EDM searches is very well motivated. Still, even with the discovery of an EDM in one system several experiments will be required to pin down the precise nature and the underlying processes. Therefore searches are going on presently in a variety of systems, ranging from free leptons to complex condensed matter samples. These experiments utilize typically state of the art precision measurements which are often based on forefront technological developments. The experimental efforts are complemented and guided by the further development and refinement of particle theory. Here a few aspects of recent developments in this exciting field are summarized.