I will discuss some recent progress in studying hadrons using first-principles lattice QCD calculations. The spectroscopy and interactions of hadrons probe the strongly-interacting regime of QCD, and in recent years experiments have observed a number of puzzling hadrons that challenge our understanding of the strong interaction. Lattice QCD provides a method for performing first-principles computations of the properties of hadrons and hence a QCD-based understanding of the phenomena. However, the majority of hadrons are unstable -- they decay strongly and appear as resonances in hadron-hadron scattering -- and lattice calculations of these are challenging. I will summarise recent advances in this area and present some applications in the light and heavy-meson sectors, including the first determination of an exotic J^PC = 1^-+ hybrid meson resonance in a lattice calculation, likely connected to the π1 observed by COMPASS, and calculations of Dπ, DK and DKbar scattering relevant for the D*0(2300) and the enigmatic D*s0(2317).