Quantum tunnelling plays an important role in dynamics and spectroscopy of molecules. It can significantly affect reactivity and allow molecular rearrangements which would otherwise be forbidden. Instanton method provides a way to approximate and visualise tunnelling dynamics using an optimal tunnelling path along which the quantum process predominantly take place. It relies on the optimization of the path, thus it enables us to treat much larger molecular systems in full dimensionality or rely on more accurate on-the-fly electronic potentials than the more accurate methods. The aim of this project is to extend the domain of applicability of the instanton method and to increase its efficiency in order to study systems that would otherwise be inaccessible. Specifically, we aim to develop a new instanton theory to calculate tunnelling splittings in vibrationally excited states and rotational energies. We will apply those theories to study tunnelling spectra in water clusters that have recently been measured, which will provide tests of water potentials at far-from-equilibrium geometries and insights into dynamics of hydrogen bonds, how they break, form and rearrange. Further applications envisage the study of the substituent effect on hydrogen bonds in some substituted carboxylic acid dimers through the associated tunnelling splittings. We also aim to develop methods for efficiently locating optimal tunnelling paths for rate calculations in nonadiabatic systems. These will be applied to study photo-excited nonadiabatic dynamics of indole or phenol and interpret time-resolved photoelectron spectra using a combination of surface-hopping dynamics above the H atom detachment barrier and instantons for the dissociation rates below the barrier. Finally, we will apply our methods to calculate tunnelling rates for selected free radical reactions which take place in aqueous solutions to determine mechanisms and estimate kinetic isotope effect.