In recent years, neuromorphic architectures have been an increasingly effective tool used to solve big data problems. Hardware neural networks have not been able to fully exploit the power efficient properties of the neural paradigm, however, due to limitations in standard CMOS. One of the largest challenges is the quadratic scaling of the synapses in a neural network. There has been some work in using post CMOS technology as synapses to overcome this limitation, but systems to date have not been scalable due to the design of their neurons. This dissertation aims to design and build scalable neural network architectures that can use emerging resistive memory technology as synapses. Using analog computing techniques to build networks is promising, especially due to the development of dense, CMOS compatible analog resistive memories. Building functional analog networks in advanced technology nodes, however, is challenging due to the relatively poor performance of analog components in these nodes. This work explores oscillatory neural networks (ONNs), which use phase as the analog state variable instead of voltage or current, reducing the number of traditional analog components required and making the networks better-suited for advanced nodes. This thesis develops additional ONN theory with regard to hardware networks, since previous work did not consider the effect of transmission delay on network dynamics. Transmission delay is proven to cause desynchronization in unmodified ONNs, and the theoretical analysis suggests ways to build networks which do synchronize. Conclusions from the theoretical development are used to build a PLL-based ONN in hardware. The PLL-based ONN is more energy efficient than comparable systems implemented in digital CMOS, although the neuron area is somewhat larger. The measurement of the PLL-based ONN also reveals additional poorly-studied facets of ONN dynamics. Using the knowledge gained from the PLL-based ONN, a larger, PLL-free ONN is built in the same technology. Removing the PLL in each neuron reduces the power and area consumption without sacrificing any functionality.This dissertation demonstrates that ONNs are well-suited to take advantage of emerging resistive memory technology to build efficient hardware neural networks.