Groupoidification is a form of categorification in which vector spaces are
replaced by groupoids, and linear operators are replaced by spans of groupoids.
We introduce this idea with a detailed exposition of "degroupoidification": a
systematic process that turns groupoids and spans into vector spaces and linear
operators. Then we present three applications of groupoidification. The first
is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator
arises naturally from degroupoidifying the groupoid of finite sets and
bijections.