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

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in ${\mathbb{R}}^3$ 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.