The mathematical modelling of cell motility naturally leads to free boundary and moving interface problems.
One example is the case when the moving cell membrane is regraded as the unknown, for example as the solution to a geometric evolution law. Another scenario of interest is when a free
boundary problem is  posed on the, possibly evolving, cell membrane itself, such models arise in cell polarisation or cell adhesion. In this talk we focus on the modelling, simulation and
analysis of such problems.  Novel mathematical and numerical methods for the analysis and simulation of geometric evolution laws, free boundary problems and partial differential equations posed
on moving interfaces will be discussed. Numerical simulations will also be presented.