The excited spectrum of hadrons is a spectrum of rapidly decaying resonances whose existence we infer from observations of their decay products. The underlying theory of quark and gluons, QCD, can be studied numerically using lattice techniques where the fields are considered on a grid of finite size. I will describe how we can make use of the finite box in which lattice QCD calculations are performed to learn something about hadron scattering amplitudes from first principles. These amplitudes contain information about the resonance structure of QCD and hence the spectrum of excited mesons and baryons. I will present the results of recent calculations in which the lightest scalar, vector and tensor mesons have been studied.