An affine-invariant property over a finite field is a property of functions over F_p^n that is closed under all affine transformations of the domain. This class of properties includes such well-known beasts as low-degree polynomials, polynomials that nontrivially factor, and functions of low spectral norm. The last few years has seen rapid progress in characterizing the affine-invariant properties which are testable with a constant number of queries. We survey the current state of this project.