In calculus, the curvature of a curve at a point in space is defined to agree with the inverse of the radius of the “tangent circle”. Menger introduced a non-local curvature that is a function of 3 points, and defines the curvature to be the inverse of the radius of the unique circle containing all 3 points. The Menger curvature can be used to study the curvature of a set that may not be differentiable. In this talk, we explore the relationship between finiteness of Menger-type curvatures in higher-dimensions, and regularity of sets. In particular, we characterize (uniform) rectifiability in arbitrary dimension and co-dimension by studying different quantities related to the generalized (integral) Menger curvature. Time allowing, I will also present an example discovered jointly with Sean McCurdy that exhibits sharpness in some other quantitative characterizations of (uniform) rectifiability.