The prospect of defining complementary logic monolithically in a wide-bandgap semiconductor such as Gallium Nitride, to manage the RF and power applications of an an energy-efficient and always-connected world, has attracted massive interest in recent years. This thesis introduces the subject with a tour of complementary logic outlooks in a range of wide-bandgap and heterogenous architectures. Then it zooms into GaN-on-AlN, the platform on which our team has been developing this possibility, and discusses mathematically the design of both n-channel and p-channel devices therein. Then the thesis demonstrates the first generation of recessed GaN/AlN p-channel devices that allowed this material system to stake its claim as a contender and discusses how to model such devices. With that flag in place, we then divert into more fundamental physics, providing a theoretical coverage of the transport possible in this high-quality platform. Finally, the thesis concludes with a set of devices acheiving current-levels in the same order of magnitude as many normally-off GaN HEMTs, a threshold which indicates that, finally, the dream of co-integration could offer true design value.