During the early development of calculus, eminent mathematicians such as Leibniz and Cauchy freely used infinitesimals in their calculations. Once the mathematical community became dubious of their status, the use of infinitesimals was replaced by the now familiar epsilon-delta rigor. In the 1960s, Abraham Robinson used techniques from model theory to rescue infinitesimals from their squalid state and instead put them on a firm foundation in what he called nonstandard analysis. Since its inception, nonstandard techniques have proven useful in many diverse areas of mathematics, from geometry to functional analysis to mathematical finance. Besides allowing one to give precise meaning to intuitive, heuristic arguments involving “ideal” elements, nonstandard analysis offers new techniques such as hyperfinite approximation and Loeb measure. In this talk, I will survey some uses of nonstandard analysis in Lie theory and additive combinatorics. Some highlights of the talk will be the nonstandard solution to Hilbert’s fifth problem (and its extension to the local case), the Breuillard-Green-Tao structure theorem for approximate groups, and some progress on a sumset conjecture of Erdos.