Monte Carlo calculations in Lattice QCD can be used to study the spectrum of quantum chromodynamics in a finite box from first principles. Most experimentally observable hadrons however appear as unstable resonances in scattering amplitudes and are hence not directly accessible through Euclidean time simulations in a finite volume. Fortunately the stationary-state energies in finite volume can be related to scattering phase shifts, hence allowing us to extract infinite-volume physics from Lattice QCD computations. These so-called Lüscher methods open exciting prospects for the study of resonant processes from first principles. In this talk I will review the stochastic LapH method for lattice spectroscopy, which enables us to systematically carry out this program. I will present results on pion-pion scattering from simulations at a single lattice spacing with $m_\pi = 240$ MeV. In the isovector channel, the large $L = 3.8$ fm volume enables good resolution of the $\rho$ resonance shape. Preliminary results from a different ensemble are also shown, including a first-principles determination of the low-energy contribution to the hadronic vacuum polarization.