Abstract: In my talk I will discuss Laplacian growth models, describing  the evolution of surfaces separating different domains, filled with different materials or different phases of the same material.   More precisely, I will introduce a generalisation of the Mullins-Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs-Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. In the second part of the talk I will introduce a new stable,  structure-preserving parametric finite element approach to the multi-phase Mullins-Sekerka problem. Numerical simulations will show qualitative properties  of solutions.