Mean curvature flow with surgery of mean convex surfaces in $$\mathbb {R}^3$$R3

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $$\mathbb {R}^3$$R3 with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author (Invent Math, 2015).