Abstract: Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give an introduction to contact geometry, an introduction to some ideas of curvature in Riemannian geometry, and discuss the iterations of the two types of geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it.  This is joint work with Rafal Komendarczyk and Patrick Massot.