Curvature is a fundamental concept in physics and it plays a crucial role in various areas such as: classical mechanics, general relativity, optics, and fluid dynamics. In particular, the curvature of surfaces can affect the mechanical, electrical, and optical properties of materials, so curvature effects need to be taken into account when designing and analyzing new materials. The recently introduced concept of nonlocal curvatures provide a frameworks for measuring the “bend” of a surface under little or no smoothness assumptions, while connecting to classical curvature as the horizon of interaction converges to zero. In this talk I will focus on two distinct problems: of constant nonlocal curvature and of ordered curvature and show how they relate to their classical counterparts.