In this thesis we summarise several results in the literature which show the approximation capabilities of multilayer feedforward artificial neural networks. We show that multilayer feedforward artificial neural networks are capable of approximating continuous and measurable functions from Rn to R to any degree of accuracy under certain conditions. In particular making use of the Stone-Weierstrass and Hahn-Banach theorems, we show that a multilayer feedforward artificial neural network can approximate any continuous function to any degree of accuracy, by using either an arbitrary squashing function or any continuous sigmoidal function for activation. Making use of the Stone-Weirstrass Theorem again, we extend these approximation capabilities of multilayer feedforward artificial neural networks to the space of measurable functions under any probability measure.