This talk is based on joint work with Xin Zhou (Cornell). I will describe our work on the existence of constant mean curvature 2-spheres. In particular, we show that in a 3-sphere equipped with a Riemannian metric of positive Ricci curvature, for all H there exists a non-trivial, branched immersed 2-sphere with constant mean curvature H. Time permitting, I will also talk about our more recent work applying some of the ideas involved in the proof of the above result to study the existence of curves with constant geodesic curvature in Riemannian 2-spheres.