Microwave parametric amplifiers based on Josephson junctions have become a key component of many quantum information experiments. One key limitation which has not been well predicted by theory is the gain saturation behavior which determines its ability to process large amplitude signals. The typical explanation for this behavior in phase-preserving amplifiers based on three-wave mixing is pump depletion, in which the consumption of pump photons to produce amplification results in a reduction in gain. In this talk, however, I will present our recent numerical and experimental studies which show that the fourth-order Kerr nonlinearities inherent in the Josephson junctions are the dominant factor of gain saturation in the Josephson Parametric Converter (JPC). I will also present our ongoing efforts to eliminate these higher order nonlinearities through Hamiltonian engineering and novel device designs.