In this talk, I will present a new mathematical formalism of geometric bulk-surface partial differential equations (G-BS-PDEs) that allows us to naturally couple biochemical processes and mechanical properties to study mechanisms for 2- and 3-D single and collective cell migration. This formalism allows us to study both intra- and extra-cellular spatiotemporal dynamics associated with the biochemical processes and mechanical properties of single cells during migration. I will present three approaches, a geometric surface partial differential equation approach whereby cell migration is driven only by forces acting on the cell plasma membrane, an image-based geometric surface partial differential equation approach where cell migration is based on an optimal control approach using phase-fields and finally, a geometric bulk-surface partial differential equation approach that couples the biochemistry of actin and myosin molecular concentrations with a visco-elastic continuum mechanical model for the actin cytoskeleton displacements. All three formalisms allow us to study how cells migrate through isotropic and non-isotropic environments as well as providing valuable biological information such as computing cell morphological changes, speed, velocity, area/volume, surface tension, curvature, proliferation rates during cell division (a process that is generally undertaken manually in experimental laboratories). Single and collective cell migration are essential for physiological, pathological and biomedical processes in development, repair, and disease; for example., in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion, and inflammation. Hence, this work contributes to experimental, biomedical, bioengineering and mechanobiology by proposing a mathematical and computational framework that naturally encodes the biophysics of cells to study mechanisms for single and collective cell migration through complex anisotropic environments. This formalism paves way for designing in-silico computational cells suitable for probing and investigating key processes associated with cell migration