Abstract: Given a foliated manifold $M$, one feels that a leafwise smoothing operator on M ought to be represented by a smooth function on the subset of $M\times M$ consisting of pairs of points lying on the same leaf. However, this subset may fail to be a manifold due to an interesting and important phenomenon known as $\textbf{holonomy}$. The $\textbf{holonomy groupoid}$ of a foliation is, roughly speaking, what results when one attempts to smooth out this singular set, somewhat in the spirit of blowups in algebraic geometry. In this talk, I will attempt to explain the concept of holonomy and sketch some work I have done on the groupoids and algebras associated to singular foliations, based on work of Androulidakis and Skandalis.