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.