Littlewood-Richardson coefficients are fundamental constants in several fields of mathematics (and in nature). In combinatorics, they appear in the ring of symmetric functions; in representation theory, they appear in the representations of groups such as GL(n) and S_n. In geometry they turn up in the topology of the Grassmannian, which parametrizes sub-vector spaces of an n-dimensional vector spaces. (This is the "geometry behind linear algebra".) I will describe how to interpret Littlewood-Richardson numbers in this way, and show you the key idea behind being able to understand them with pictures (the "geometric Littlewood-Richardson rule"). I'll conclude with a list of applications in several fields, but the main goal of this talk will be to communicate the flavor of the ideas involved. In particular, no background will be assumed.