In spite of much progress, our basic understanding of mean curvature flow is in some respects still lacking, apart from the case of curves in the plane. However, beautiful work of White and Huisken-Sinestrari in the last 10 years has shown that there is a far-reaching structure and regularity theory in the case of mean convex (i.e. positive mean curvature) mean curvature flow. After presenting some background, I will discuss joint work with Robert Haslhofer, which gives a new approach to mean convex flow that is substantially simpler and shorter than the original.