The Circle of Willis (CoW) is a ring-like structure of blood vessels found at the base of the brain. Its main function is to distribute a constant flow oxygen-rich arterial blood to the cerebral mass, despite changes in afferent pressures or flows. This objective is achieved by a local mechanism known as autoregulation, whereby the resistance in small vessels branching from the CoW changes by vasodilation or vasoconstriction of the smooth muscle cells surrounding the vessel. A one-dimensional (1D) model of the CoW is developed to simulate a series of possible clinical scenarios such as occlusions in afferent arteries, absent or string-like circulus vessels, or arterial infarctions. A series of studies investigates various features of autoregulatory behaviour. Firstly, a simple model is created to verify solution methods; secondly, the model is validated against a three-dimensional (3D) Computational Fluid Dynamics (CFD) model; and lastly, the decentralised nature of cerebral autoregulation is investigated. Finally, an advanced, metabolic model of autoregulation is created, incorporating the successful aspects of the early model, as well as more physiologically accurate dynamics. The advanced model captures cerebral haemodynamic autoregulation by using a Proportional-Integral-Derivative (PID) controller to modify efferent artery resistances and partial pressures of oxygen to maintain optimal efferent flow rates and oxygen supply to the cerebral mass for a given circle geometry and afferent blood pressure. This advanced model is physiologically relevant, matching the accepted physiological responses of blood flow as a function of arterial pressure, tissue oxygen partial pressure as a function of blood flow, as well as limited transient clinical data. Results match accepted physiological response and exhibit excellent correlation with the limited clinical data available. In addition, a set of boundary conditions and geometry is presented for which the autoregulated system cannot provide the necessary efferent flow rates and perfusion, representing a condition with increased risk of stroke and highlighting the importance of modelling the haemodynamics of the Circle of Willis. The system model created is computationally simple so it can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures. In addition, the solution for the CoW arterial system is obtained in a far shorter time period using this time-varying resistance model than with higher dimensional CFD methods, and requires significantly less computational effort while retaining a high level of accuracy.