Neural networks can very effectively perform multidimensional nonlinear classification. However, electronic networks suffer from significant bandwidth limitations due to carrier lifetimes and capacitive coupling. This project investigates photonic neural networks that can get around these limitations by performing both the activation function and weighted addition in the optical domain using microring resonators. These optical microring resonators provide both nonlinearity and superior fan-in without compromising bandwidth. The ability to thermally calibrate networks of cascaded axons and dendrites and train such a network to solve nonlinear classification problems are demonstrated using theory and simulations. The former is also demonstrated experimentally on a two-channel axon cascaded into a two-channel dendrite, showing good agreement between simulation and experiment. In addition, the use of transverse modes to increase the size of each photonic layer is examined. Simulations that determined the optimal waveguide geometry for using these modes were experimentally validated.First and foremost, I would like to express my deepest gratitude to Alex Tait 'GS, my primary day-to-day mentor and advisor. Without him, absolutely none of this work would have been possible. I would also like to extend my thanks to the entire Lightwave Communications Laboratory for their advice, support, and infrastructure. This project was also made possible by generous funding from the School of Enginering and Applied Science and the Department of Electrical Engineering. Finally, I would like to acknowledge all of my friends and peers: the fighting ELE Class of 2017, off which I could always bounce ideas, and my friends and roommates, who were willing to listen to my ramblings and help edit this document. Thank you to everybody for helping make this a success.This work is dedicated to the loving memory of my parents, Lauren and David Gordon.