Abstract: The notion of an infinitesimal quantity has been used in mathematics for over 2200 years. It eluded rigorous treatment until the work of Abraham Robinson in 1960 established a rigorous foundation for the use of infinitesimals in mathematics. Recent extensions and applications of his theory, called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. In all of these areas, infinitely small and infinitely large quantities can play an essential role in the creative process. At the level of calculus, the integral can now be correctly defined as the nearest ordinary number to a sum of infinitesimal quantities. In Probability theory, Brownian motion can now be rigorously parameterized by a random walk with infinitesimal increments. In economics, an ideal economy can be formed from an infinite number of agents, each having an infinitesimal influence on the economy. The talk will give an overview of the basis for this very old, yet relatively new, area of mathematics.