In 1997, Maxim Kontsevich gave a universal formula for the quantization of Poisson brackets. It is an example of a perturbative expansion in quantum field theory, and as such, its coefficients are given by complicated high-dimensional integrals associated to graphs. I will describe forthcoming joint work with Peter Banks and Erik Panzer, in which we prove that these integrals can be expressed as integer-linear combinations of special transcendental constants: the even-weight multiple zeta values. Our proof builds on Francis Brown's approach to the periods of the moduli space of genus zero curves and Oliver Schnetz's technique of single-valued integration for polylogatithms. It yields the first general algorithm and software for the symbolic calculation of the quantization formula.