Atomistic simulations are a bottom up approach that predict properties of materials by modelling the quantum mechanical behaviour of all electrons and nuclei present in a system. These simulations, however, routinely assume nuclei to be classical particles, which leads to incorrect predictions for systems that exhibit significant quantum delocalization and zero-point effects, such as those containing light nuclei. The path integral approach, the state of the art approach that models the exact quantum thermodynamics of this class of systems, is much more computationally expensive and harder to implement than the classical methods that evolves the system using classical statistical mechanics. This has prevented widespread modelling of the quantum mechanics of nuclei in atomistic simulations, especially in combination with computationally expensive interatomic potentialsthat model interparticle interactions at a high level of theory. In this thesis, we present several new methods that dramatically reduce the computational cost of modelling the quantum nature of nuclei with respect to standard methods, and to existing cost reduction schemes. These methods are based on the realization that nuclear quantum effects can often be modelled using cheap short ranged interaction potential, or using high order splittings that decouple non-commuting potential and kinetic energy operators, or using generalized Langevin equations that can be used to mimic quantum fluctuations with correlated noise. We have also derived bespoke estimators of quantities such as the quantum heat capacity, the particle momentum distribution, and vibrational spectra that reduce the cost of calculating these properties, and allow direct comparisons with experiments. These methods have been implemented in the second release of an open source software i-PI, which allows them to be used in combination with widely used softwares that compute interatomic potentials. The availability of these methods has promoted routine incorporation of nuclear quantum effects in atomistic simulations. The relevance of these advances is underscored by the different properties and classes of materials to which we have applied these methods. For instance, we have computed PMD in different phases of water facilitating interpretation of Deep Inelastic Neutron Scattering experiments, and the understanding of the local environments of protons. Similarly, we have shown how the interplay of quantum effects and intermolecular interactions can be used to tune the heat capacity of methane loaded metal-organic frameworks, to increase, decrease or stay constant over a range of temperatures. We have also studied the impact of NQEs in affecting stabilities of pharmaceutically active molecular crystals using several computationally inexpensive methods that are routinely used to approximate quantum free energies. We have systematically studied their accuracy on a large set of solids, and concluded that free energy calculations that include the quantum nuclear motion exactly are the only reliable by way of predicting stabilities of molecular crystals.