We review a notion of stability for graded algebras, which gives rise to a non-commutative generalization of Hilbert stability for projective varieties. We report on recent work, joint with Junho Hwang, which describes the first examples of stable algebras. Namely, we determine which quadratic regular algebras of dimension 3, i.e., non-commutative projective planes, are stable. We describe the moduli stack of stable non-commutative projective planes. It has 4 components, corresponding to Types A, B, E, and H in the Artin-Schelter classification of such algebras.