Entanglement in periodically driven quantum systems Zusammenfassung: The entanglement of quantum states is one of the most fundamental aspects of quantum theory that has led to fertile debates and practical applications. In the present thesis we investigate how entanglement evolves in the presence of a periodic driving force. Our motivation are recent findings in the literature which indicate that periodic driving can render entanglement more robust against the detrimental influence of the environment which is present in any realistic quantum system. In particular we are interested in identifying the conditions for the periodic drive to sustain entanglement in the limit of long times i.e. in the cyclo stationary state. This question is discussed from different perspectives: After an introduction and a summary of the theoretical concepts in Chapter 1 and 2 Chapter 3 has its focus on periodically driven quantum systems in which the influence of the environment can be neglected. There we investigate the entanglement properties of the resulting Floquet states which define in essence periodic solutions of the dynamics. In Chapter 4 we include environment induced decoherence and dissipation mechanisms in our description and analyze the entanglement of the resulting asymptotic cycle. In this context we develop a very general approach to determine the optimal periodic driving mechanism for a given target quantity (which can be inter alia entanglement)