The nuclear dynamics of molecules following photoexcitation are fundamentally quantum dynamical processes owing to the breakdown of the Born-Oppenheimer approximation in regions close to the intersection of potential energy surfaces. To describe such processes, the solution of the time-dependent Schrödinger equation is necessitated. If large amplitude nuclear motion is involved, then this is a particularly challenging task for conventional gridbased quantum dynamics methods, owing to the need to construct model Hamiltonians that accurately describe multiple potential energy surfaces and the couplings between them over a large subvolume of nuclear configuration space. A powerful alternative is to expand the nuclear wavefunction in terms of localised time-dependent parameterised basis functions, and to exploit this locality to calculate Hamiltonian matrix elements using information at a small number of nuclear geometries. Such methods not only promise to break the curse of exponential scaling suffered by conventional grid-based methods, but also allow for quantum dynamics calculations to be performed ‘on-the-fly’ using information from ab initio electronic structure calculations calculated as and when it is needed. In this talk, I will discuss one such method: the ab initio multiple spawning (AIMS) method. In the AIMS method, the solution of the time-dependent Schrödinger equation is achieved via the expansion of the nuclear wavefunction in an adaptive set of Gaussian basis functions, which is increased in size when needed in order to efficiently describe the transfer of population between electronic states. In the first part of the talk, the theoretical foundations and details of the AIMS method will be discussed. In the second part, representative examples of the study of excited state molecular dynamics will be given, illustrating the power and success of the AIMS method in studying photochemical processes.