abstract: In this talk, I will present a series of mathematical models for 3D cell migration which includes geometric partial differential equations, coupled bulk-surface models and mechanochemical models. These models describe the coupling of biochemical reactions with biomechanical properties which are responsible for cell migration through homogeneous environments. Novel numerical methods will be presented to tackle the different models. Numerical simulations will be presented that support the theoretical modelling framework with applications to real world problems for cell migration. Cell migration is a multistep process essential for mammalian organisms and is closely linked to processes such as development, immune response, wound healing, tissue differentiation and regeneration, inflammation, tumour invasion and metastasis formation.