This paper reviews the use of an exact renormalization group equation in quantum field theory and statistical physics, focusing on the dependence of the free energy on an infrared cutoff for quantum or thermal fluctuations. Non-perturbative solutions are derived from approximations to the coarse-grained free energy or effective average action, which interpolate between microphysical laws and complex macroscopic phenomena. The approach is applied to $O(N)$-symmetric scalar models in various dimensions, covering critical phenomena, including the Kosterlitz-Thouless transition and critical behavior of polymer chains. Universal aspects of critical equations of state are computed, and a direct connection between microphysical and critical quantities for a liquid-gas transition is established. The paper also discusses first-order phase transitions in scalar matrix models, the quantitative treatment of coarse graining for nucleation rates, quantum statistics in thermal equilibrium, and high-temperature symmetry restoration in theories with spontaneous symmetry breaking. Particular attention is given to chiral symmetry breaking and the high-temperature or high-density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.This paper reviews the use of an exact renormalization group equation in quantum field theory and statistical physics, focusing on the dependence of the free energy on an infrared cutoff for quantum or thermal fluctuations. Non-perturbative solutions are derived from approximations to the coarse-grained free energy or effective average action, which interpolate between microphysical laws and complex macroscopic phenomena. The approach is applied to $O(N)$-symmetric scalar models in various dimensions, covering critical phenomena, including the Kosterlitz-Thouless transition and critical behavior of polymer chains. Universal aspects of critical equations of state are computed, and a direct connection between microphysical and critical quantities for a liquid-gas transition is established. The paper also discusses first-order phase transitions in scalar matrix models, the quantitative treatment of coarse graining for nucleation rates, quantum statistics in thermal equilibrium, and high-temperature symmetry restoration in theories with spontaneous symmetry breaking. Particular attention is given to chiral symmetry breaking and the high-temperature or high-density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.