Title: The geometry of constant mean curvature surfaces embedded in R^3 Abstract: In this talk I will discuss recent results on the geometry of constant mean curvature (H\neq 0) surfaces embedded in R^3. Among other things I will prove radius and curvature estimates for simply connected surfaces embedded in R^3 with constant mean curvature. It follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.