This talk will be a survey of problems and results about the volume of certain geometrically interesting subsets of Euclidean space. For example, we will describe Gromov’s waist inequality for spheres, and Gromov's conjectured waist inequality for cubes. The waist inequality for cubes would be a deep non-linear generalization of the cube slicing inequality proved by the speaker in 1978. Another interesting problem is the volume of the set of points in Euclidean space which are the coefficients of a polynomial with Mahler measure bounded by 1. This set is the unit ball for a non-convex distance function, and turns out to have a volume which is a rational number in each dimension. As time permits, several further problems and results about the volume of interesting sets will be described. The talk will be mostly expository, and suitable for a general mathematical audience.