Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.