Modern hadron spectroscopy on the lattice: From bound state masses to scattering amplitudes

While the masses of stable hadrons are routinely calculated in Lattice QCD, the determination of resonance properties with controlled systematic uncertainties poses considerable challenges, in particular for mesons containing heavy quarks. After some general introduction, I will use examples from recent and current Lattice QCD simulations to highlight the progress and difficulties in determining the hadron spectrum of QCD from first principles. To start, results for hadrons stable under QCD from a precision calculation of mass splittings in the charmonium spectrum will be presented. The procedure for and impact of resonance physics will then be highlighted on the simple example of the rho-meson. In this case, the precise determination of the finite volume spectrum and the timelike pion form factor at low energies improves the reliability and accuracy of the lattice calculation of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. Finally, I will discuss some exploratory results for QCD bound states and resonances in the heavy-light and charmonium spectra, where the ultimate objective is a better understanding of the unexpected (and likely exotic) X, Y and Z mesons.