This thesis deals with the development and analysis of neural fields and neural networks. Neural fields model the averaged scale behaviour of large groups of neurons, where we include a transmission delay and a diffusion term modelling gap junctions. We investigate the dynamical behaviour of such models and study Hopf bifurcation with and without symmetry. We also expand the mathematical framework for delay equations, the sun-star calculus, to cover our model. Neural networks are used in machine learning to learn arbitrary mappings on some data set. We investigate these networks in a continuum limit using functional analysis and show how they fit in the reproducing kernel Banach spaces.