Neurodegenerative diseases pose a major challenge for contemporary medicine, driven by the rapid aging of the global population. A central class of these disorders is the family of proteinopathies (i.e., Alzheimer's and Parkinson's diseases), where disease onset is linked to the misfolding, spreading, and aggregation of proteins in the brain. Mathematical modelling plays a key role in describing disease progression and in supporting clinical practice, by capturing the dynamics of prion-like proteins and the response of brain physiological systems, while advanced numerical methods are required to simulate these processes in silico. In this talk, I will present mathematical models and numerical schemes for simulating neurodegenerative diseases in realistic brain geometries. The discretization is based on high-order discontinuous Galerkin methods on polygonal and polyhedral meshes, which are particularly well suited to resolve the complex brain surface and internal interfaces with high geometric fidelity. Special attention is devoted to structure-preserving formulations, in order to guarantee physically meaningful concentrations of misfolded proteins. Finally, I will show how biological post-mortem data can be exploited to inform and validate the models. The computational framework is implemented in lymph, a new software library for multiphysics simulations based on polygonal discontinuous Galerkin methods.