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The outcome of chemical reactions proceeding via radical pair (RP) intermediates can be influenced by the magnitude and direction of applied magnetic fields, even for interaction strengths far smaller than the thermal energy. Sensitivity to Earth-strength magnetic fields has been suggested as a biophysical mechanism of animal magnetoreception and this thesis is concerned with simulations of the effects of such weak magnetic fields on RP reaction yields. State-space restriction techniques previously used in the simulation of NMR spectra are here applied to RPs. Methods for improving the efficiency of Liouville-space spin dynamics calculations are presented along with a procedure to form operators directly into a reduced state-space. These are implemented in the spin dynamics software Spinach. Entanglement is shown to be a crucial ingredient for the observation of a low field effect on RP reaction yields in some cases. It is also observed that many chemically plausible initial states possess an inherent directionality which may be a useful source of anisotropy in RP reactions. The nature of the radical species involved in magnetoreception is investigated theoretically. It has been shown that European Robins are disorientated by weak radio-frequency (RF) fields at the frequency corresponding to the Zeeman splitting of a free electron. The potential role of superoxide and dioxygen is investigated and the anisotropic reaction yield in the presence of a RF-field, without a static field, is calculated. Magnetic field effect data for Escherichia coli photolyase and Arabidopsis thaliana cryptochrome 1, both expected to be magnetically sensitive, are satisfactorily modelled only when singlet-triplet dephasing is included. With a view to increasing the reaction yield anisotropy of a RP magnetoreceptor, a brief study of the amplification of the magnetic field experienced by a RP from nearby magnetite particles is presented. Finally in a digression from RPs, Spinach is used to determine the states expected to be immune from relaxation and therefore long-lived in NMR experiments on multi-spin systems.