Fedosov and, independently, de Wilde and Lecomte proved that every smooth symplectic manifold admits a deformation quantization. Bezrukavnikov and Kaledin later established an analogous result for affine symplectic varieties over a field of characteristic zero.

Although the quantized algebra \mathcal{O}_h is not canonically defined, the category of \mathcal{O}_h-modules is. In particular, a canonical categorical quantization exists for any symplectic variety over a field of characteristic zero.

I will review some of these results and then explain an analogous picture in positive characteristic (due to Bezrukavnikov-Kaledin and Bogdanova-Kubrak-Travkin-Vologodsky). A new feature in characteristic p is that the categorical quantization is defined over a mod-p analogue of the twistor line rather than merely over a formal disk.

If time permits, I will also discuss a construction, due to Travkin, of the canonical categorical quantization over the ring of integers.