Curvature-dimension inequalities are modifications of Ricci curvature bound or, in the language of relativity, an energy condition. They have proved useful in applications of Fourier analysis to diffusion processes. Surprisingly (to me), as tools to prove theorems in Riemannian geometry and general relativity, they are as powerful as the usual Ricci curvature bounds and can yield new results. Applications include static Einstein metrics, near-extremal-horizon geometry, and scalar-tensor gravity. I will discuss these applications and use curvature-dimension bounds to prove some singularity theorems and Lorentzian splitting theorems.