We present recent work, joint with J. Wolf, in which we define a natural notion of higher-order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic subvarieties up to linear error. This generalizes previous joint work with Wolf on arithmetic regularity lemmas for stable subsets of $\mathbb{F}_p^n$ to the realm of higher-order Fourier analysis.