Often the complex shapes of living matter and the dynamics on their surfaces cannot be accurately described by partial differential equations (PDEs) in simple domains (bulk). This challenge can be approached with the aid of geometric PDEs. In the first part of this talk we present a brief overview about recent advances in the area of PDEs on surfaces and their coupling with PDEs in the bulk. Herein we mainly focus on the numerical challenges thereof and prospective applications in cell migration or tumor growth. The second part is devoted to our current work on a stabilized finite element solver for PDEs on evolving surfaces with bulk processes. We employ the level set methodology to implicitely define the sharp surface and treat convection-dominated flow with an algebraic flux correction technique.