In their paper "Construction of Instantons," Atiyah, Drinfeld, Hitchin, and Manin introduced an algebraic construction of the moduli space of instantons on R^4, now also known as the "ADHM space." This is a Poisson complex variety; it has been actively studied by both mathematicians and physicists. In this talk, I will review the ADHM construction, present examples, and discuss various geometric and algebraic properties of ADHM spaces. I will also describe natural quantizations of these Poisson varieties. I will explain a joint result with Etingof, Losev, and Simental, providing explicit formulas for the dimensions and characters of all finite-dimensional representations of these quantizations. Time permitting, I will discuss an application of our result to combinatorics and the connection of this story to knot invariants.