In this talk, I will present our most recent results  based on two finite element formulations: (i) the surface finite element and (ii) the projected finite element methods applied to solving partial differential equations of reaction-diffusion type on arbitrary stationary and evolving surfaces.  Reaction-diffusion equations on evolving surfaces are formulated using the  material transport formula, surface gradients and diffusive conservation laws.
The evolution of the surface is defined by a material surface velocity. The projected finite element method differs from the surface finite element method in that it provides a conforming finite element discretization which is "logically" rectangular. However, this property restricts the general applicability of the numerical method to arbitrary surfaces, a key advantage for the evolving surface finite element method.  To demonstrate the capability, flexibility, versatility and generality of the numerical methodologies proposed, I will present various numerical results. This methodology provides a framework for solving partial differential systems on continuously evolving surfaces. Reaction-diffusion models have numerous applications in developmental biology, cancer research, wound healing, tissue regeneration, and cell motility.