In 1997, Kontsevich constructed a universal quantization of
every Poisson manifold as a formal power series. Its coefficients are given as integrals over moduli spaces of marked holomorphic discs. In joint work with Peter Banks and Brent Pym, we show that these integrals always evaluate to multiple zeta values, which are very well studied transcendental numbers. I will give an introduction to deformation quantization, explain Kontsevich's formula and our result and discuss the main ideas of the proof.