The family of copper-oxides collectively known as the "cuprates" exhibits the remarkable phenomenon of high-temperature superconductivity (high-Tc). In some cuprates this macroscopic quantum-mechanical state, with the zero electrical resistance, extends more than half way to room temperature (164 K). Despite almost 30 years of intensive research in this field---research that has driven fantastic developments in techniques like scanning-tunneling-microscopy, angle-resolved photoemission, and neutron and x-ray scattering, to name a few---the mechanism of high-Tc is still unknown, and the mere "factor of two" that we need for room-temperature superconductivity has remained elusive. We use magnetic fields of up to 100 tesla to suppress superconductivity and examine the properties of the underlying metal that gives rise to high-Tc. I will review how magnetic fields give Fermi surface information, telling us about broken symmetries and interactions in the system. I will then present my latest research that uncovers the signature of quantum criticality---where ground-state quantum fluctuations dominate the interactions in a system---in the high-Tc cuprate YBa2Cu3O6+x, and discuss the relevance of quantum criticality to superconductivity. I will conclude with a discussion of future directions for high-Tc research, including what else we can learn in high magnetic fields, and ultrasonic measurements that can reveal broken symmetries in the ground-state.