This book sheds new light on the dynamical behaviour of electron spins in molecules containing two unpaired electrons (i.e. a radical pair). The quantum dynamics of these spins are made complicated by the interaction between the electrons and the many nuclear spins of the molecule; they are intractable using analytical techniques, and a naive numerical diagonalization is not remotely possible using current computational resources. Hence, this book presents a new method for obtaining the exact quantum-mechanical dynamics of radical pairs with a modest number of nuclear spins. Readers will learn how a calculation that would take 13 years using conventional wavepacket propagation can now be done in 1 day, and will also discover a new semiclassical method for approximating the dynamics in the presence of many nuclear spins. The new methods covered in this book are shown to provide significant insights into three topical and diverse areas: charge recombination in molecular wires (which can be used in artificially mimicking photosynthesis), magnetoelectroluminescence in organic light-emitting diodes, and avian magnetoreception (how birds sense the Earth's magnetic field in order to navigate).