Measurement induced geometric phases Abstract: In this thesis we combine the idea of geometric phases in quantum mechanics with the notion of weak measurements in the sense of measurements with small back action. For this purpose we examine two different measurement methods: Firstly dynamics induced by the interaction with a binary detector system which could describe for instance measurements with qubit like detectors such as two level atoms. Secondly dynamics due to continuous measurements which describe average measurement schemes like for instance electric currents flowing through quantum systems. The back action of the detector on the system modifies the state of the system. It follows that a number of measurements along different directions in the space of quantum states may drag the system along a certain quantum trajectory. If this trajectory is closed a geometric phase may be assigned to the evolution where the former is proportional to the solid angle enclosed by the trajectory on the Bloch sphere. First we show how this measurement induced geometric phase can be observed in an appropriate interference experiment. In the case of the binary measurement model we find an analytic dependence of the geometric phase on the measurement strength for one chosen realization of the measurements. For the continuous measurement model we use Monte Carlo simulations and find a highly fluctuating geometric phase in dependence of the measurement strength. Although for weak measurements the back action is small there are large fluctuations in the induced trajectories and consequently also in the geometric phases. We use different deterministic approaches to obtain analytical approximations for the average and most probable geometric phase for each measurement strength. Finally we discuss the phenomenon of geometric dephasing and show that it vanishes in the adiabatic limit