The majority of hadrons are unstable; they decay strongly to two or more lighter hadrons and appear as resonances, complex singularities in hadron-hadron scattering amplitudes. Moreover, over the last decade experiments have observed a number of puzzling states near or above thresholds. Lattice QCD provides a method to perform first-principles calculations of these phenomena that can be compared against experiment, but studying resonances and scattering is challenging. I summarise how we have recently made some significant advances in this area by using a range of techniques and discuss a couple of applications: the rho resonance in elastic pi pi scattering and coupled-channel strangeness = 1 scattering (pi K, eta K) where we found a variety of interesting phenomena (narrow and broad resonances, a bound state and a 'virtual bound state'). I comment on future prospects and further applications.