The Brascamp-Lieb inequality provides a natural generalisation of the multilinear Hölder inequality, the Young convolution inequality and the Loomis-Whitney inequality. For a number of reasons, including applications to harmonic analysis and PDE, it is of interest to establish a nonlinear variant of the Brascamp-Lieb inequality. I will explain some recent work with Jon Bennett, Stefan Buschenhenke, Michael Cowling and Taryn Flock, where we establish such a nonlinear Brascamp-Lieb inequality in full generality.