I will discuss recent progress and open challenges in computing multi-hadron electroweak amplitudes from lattice QCD, a Euclidean, finite-volume, discretised formulation of quantum chromodynamics. Starting from the underlying quarks and gluons, QCD predicts a spectrum of hadrons—such as the pion, kaon, and D meson—whose masses and matrix elements can now be computed with sub-percent precision using numerical lattice QCD. Looking beyond this, an important frontier involves scattering processes with multiple hadrons, including πK → πK, ππK, electroweak transitions such as πγ → ππ, K → ππ, D → ππ, and B → K*, and neutral-meson mixing amplitudes, as well as other long-distance matrix elements. Such observables are essential for interpreting experiments in flavour physics and testing the Standard Model. In this colloquium, I will aim to outline the key theoretical challenges in such calculations, review the methods developed to address them, and highlight a selection of recent results.

This talk is part also of the Charming Lattice QCD at CERN 2025 workshop