The thresholding scheme, a time discretization for mean curvature flow, was introduced by Merriman, Bence and Osher. In the talk I present new convergence results for modern variants of this scheme, in particular in the multi-phase case with arbitrary surface tensions. The main result establishes convergence towards a weak formulation of (multi-phase) mean curvature flow in the BV-framework of sets of finite perimeter. The proofs are based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu and Otto. This interpretation means that thresholding preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. Our proofs are similar in spirit to the convergence result of Luckhaus and Sturzenhecker, which establishes convergence of the minimizing movements scheme introduced by Almgren, Taylor and Wang, and Luckhaus and Sturzenhecker. Like theirs, also ours is a conditional convergence result, which means that we assume the time-integrated energies to converge to those of the limit. This is a natural assumption, which is however not ensured by the compactness coming from the basic estimates. We show that the methods can incorporate external forces and a volume constraint. Furthermore, I will present a similar result for the vector-valued Allen-Cahn Equation. This talk is based on joint works with Felix Otto (MPI MIS Leipzig), Thilo Simon (MPI MIS Leipzig) and Drew Swartz (Booz Allen Hamilton) Speaker(s): Tim Laux (Max Planck Institute, Leipzig)