Multiphase mean curvature flow is a widely studied system of geometric evolution equations. In the talk we introduce a notion of weak solution based on De Giorgi's general framework of gradient flows. In the first part we will recall the classical MBO scheme for mean curvature flow, we will recall the abstract framework of gradient flows in detail and we will show how it can be used to prove convergence of the classical MBO scheme to a De Giorgi solution. In the second part of the talk we introduce a new variant of the MBO scheme, due to Esedoglu and Salvador, which allows to handle general mobilities and surface tensions. In the same way as in the first part of the talk, we will be able to put this scheme in the general framework of gradient flows. Using a careful localization argument we will prove the convergence of the modified scheme to a De Giorgi solution.