Biological ion channels are responsible for, and regulate the communication system in the body. In this thesis I develop, test and apply theoretical models of ion channels, that can relate their structure to their functional properties. Brownian dynamics simulations are introduced, in which the motions of individual ions are simulated as they move through the channel and in baths attached to each end. The techniques for setting boundary conditions which maintain ion concentrations in the baths and provide a driving potential are tested. Provided the bath size is large enough, all boundary conditions studied yield the same results. ¶ Continuum theories of electrolytes have previously been used to study ion permeation. However, I show that these continuum models do not accurately reproduce the physics taking place inside ion channels by directly comparing the results of both equilibrium Poisson-Boltzmann theory, and non-equilibrium Poisson-Nernst-Planck theory to simulations. In both cases spurious shielding effects are found to cancel out forces that play an important role in ion permeation. In particular, the `reaction field' created by the ion entering the narrow channel is underestimated. Attempts to correct these problems by adding extra force terms to account for this reaction field also fail. ¶ A model of the L-type calcium channel is presented and studied using Brownian dynamics simulations and electrostatic calculations. The mechanisms of permeation and selectivity are explained as the result of simple electrostatic interactions between ions and the fixed charges in the protein. The complex conductance properties of the channel, including the current-voltage and current-concentration relationships, the anomalous mole fraction behaviour between sodium and calcium ions, the attenuation of calcium currents by monovalent ions and the effects of mutating glutamate residues, are all reproduced. ¶ Finally, the simulation and electrostatic calculation methods are used to study the gramicidin A channel. It is found that the continuum electrostatic calculations break down in this narrow channel, as the concept of applying a uniform dielectric constant is not accurate in this situation. Thus, the permeation properties of the channel are examined using Brownian dynamics simulations without electrostatic calculations. Future applications and improvements of the Brownian dynamics simulation technique are also described.