A practical application of our research is to understand how thermal fluctuations might erase magnetic memory when the recorded bits reach the scale of nanometers. Another application is to explain how nanometer-sized hot spots occur inside bulk materials. Several experimental techniques have shown that these thermal fluctuations are localized, uncorrelated with neighboring fluctuations, thereby deviating from standard thermodynamics that requires an effectively infinite and homogeneous heat bath. In 1878 Gibbs introduced the chemical potential, which accommodates the thermal energy of individual particles. In 1962 Hill introduced the subdivision potential, which accommodates the thermal energy of individual fluctuations. We find that Hill’s subdivision potential is essential to ensure conservation of energy and maximum entropy during equilibrium fluctuations. We use this “nanothermodynamics” as a guide to develop experiments, theories, and computer simulations. Experiments that we pioneered include: ultrafast SQUID magnetometry, time-domain dielectric spectroscopy, nonresonant-spectral hole burning, vertical-cantilever force microscopy, and tickle-field electron microscopy. Theories that we develop utilize Hill’s fully-open nanocanonical ensemble, yielding a mesocopic mean-field theory and local Landau theory for phase transitions. Computer simulations that we investigate include nonlinear corrections to the total energy from changes in local entropy applied to the Ising model, Creutz model, and molecular dynamics. We have shown that nanothermodynamics provides a fundamental foundation for several formulas that have been known empirically for many years, including stretched-exponential relaxation (1854), super-Arrhenius activation (1921), non-classical corrections to critical scaling (1893), and 1/f noise (1925). The fundamental goal of our research is to understand these empirical formulas, including commonly measured deviations, using nanoscale corrections to classical thermodynamics and statistical mechanics.