Could we still do geometry if the coordinates of space did not commute under multiplication? Can we imagine a world where it is impossible to know the $x$ and $y$ positions of an object simultaneously to arbitrary precision? How can we generalize the algebra that arises in the study of geometry? These questions lead to a fascinating subject called noncommutative geometry in which constructions from noncommutative algebra are interpreted spatially. In this talk, I will discuss the mathematical and physical motivations for noncommutative geometry and show how some of the technical machinery of ordinary geometry, namely the theory of differential forms and de Rham cohomology, carries over to the noncommutative world via a construction known as Hochschild homology.