Condensed Matter Physics at Extreme Conditions: High B, and Low T* Alex H. Lacerda 1, 2 National High Magnetic Field Laboratory 1 National High Magnetic Field Laboratory, Pulsed Field Facility, Los Alamos National Laboratory 2 National High Magnetic Field Laboratory, Florida State University Magnetic fields along with temperature and pressure are the most important thermodynamic parameters used to probe matter. The National High Magnetic Field Laboratory (NHMFL) mission is to maintain a strong research program and an international users program centered on the strongest magnetic fields available. Pressure to 40kBar, DC to terahertz spectroscopy, and 300K to well above ambient temperatures and more provide necessary tools for thorough exploitation of our unique magnet resources. Combining structure and phonon studies via elastic and inelastic neutron and light scattering round out a powerful suite of capabilities in use, or under development at the NHMFL. The three sites of the NHMFL located at Florida State University (Tallahassee), University of Florida (Gainesville), and Los Alamos National Laboratory (LANL) comprise the primary high field user facility of the United States. The current experimental portfolio includes DC magnetic fields to 45T (hybrid magnet) and 33T (resistive), pulsed magnets to 70T, and a newly commissioned 900MHz NMR magnet. We focus here on current and future plans for high-magnet-field technology and recent scientific highlights. The international user program at the NHMFL continues to grow, assisted by highly talented in-house scientists, new laboratory infrastructure, and the continuing development of experimental techniques to measure physical quantities rarely or never before measured in extremely high magnetic fields. Materials' strengths have limited the development worldwide of high magnetic fields, particularly short-pulse magnets beyond the 80T range. The NHMFL continues to pursue magnet engineering of single-shot destructive magnets and enhancement of routinely-available peak field and repetition rate of multi-shot magnets. Recently we achieved 75T in a short pulse duplex magnet (15mm bore, 10ms). This particular magnet is part of our effort to develop a 100T non-destructive magnet. During the last couple of years an emerging theme relates to understanding magnetically-induced new phases of condensed matter. Using neutrons (and soon, x-rays) combined with sound-speed measurements to determine phonons and structure has, in conjunction with NHMFL magnetic fields, provided new insights into plutonium science as well as many other materials. We also discuss lattice contributions to quantum criticality effects via Grunesisen physics and explore recent results in high Tc superconductors and strongly correlated materials. * The NHMFL is supported by the National Science Foundation, the US Department of Energy, and the State of Florida