Abstract: We will discuss highly precise measurements of the fine-structure constant α, which provide stringent tests of the standard model of particle physics and set strong limits on the existence of speculative particles. Our 2018 measurement used an atom interferometer to achieve an accuracy of 0.2 ppb (parts per billion); we are now setting up its successor, aiming at a tenfold improvement by taking unprecedented control over systematic effects from laser wavefront aberrations. Comparing experiment with theory relating α to the gyromagnetic anomaly g - 2 of the electron is sensitive to a broad range of previously unmeasurable physics, both within the Standard Model (such as tests of quantum electrodynamics and lattice quantum chromodynamics) and beyond (such as constituents of the dark sector or a possible substructure of the electron). Methods developed in the project will enable ultra-clean and ultra-bright optical lattices, opening up new avenues in quantum information science and precision measurement. New results - the 2020 Paris measurement of α at 80 ppt accuracy as well as the new results of the Muon g - 2 collaboration - provide strong, further motivation for continued improvements in measuring α.