In this talk I will discuss recent results on the geometry of constant mean curvature ($H\neq 0$) surfaces embedded in $\mathbb{R}^3$. Among other things I will prove a radius and curvature estimates for constant mean curvature disks embedded in $\mathbb{R}^3$. It follows from the radius estimate that the only complete constant mean curvature disk embedded in $\mathbb{R}^3$ is the round sphere.
This is joint work with Bill Meeks.