This review presents the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for quantum or thermal fluctuations. Non-perturbative solutions are obtained from approximations to the general form of the coarse-grained free energy or effective average action. These solutions interpolate between microphysical laws and complex macroscopic phenomena. The approach provides a unified description of O(N)-symmetric scalar models in two, three, or four dimensions, covering critical phenomena for second-order phase transitions, including the Kosterlitz-Thouless transition and polymer chain behavior. The critical equation of state is computed, establishing a connection between microphysical and critical quantities for liquid-gas transitions. Universal features of first-order phase transitions are studied in scalar matrix models. The quantitative treatment of coarse graining is essential for estimating nucleation rates. Quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons are discussed, as well as high-temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. The work explores chiral symmetry breaking and high-temperature or high-density chiral phase transitions in quantum chromodynamics using effective four-fermion interactions. The review discusses the non-perturbative flow equation, its solutions, and applications to scalar models, matrix models, and phase transitions. It emphasizes the importance of the effective average action in non-perturbative approximations and the role of renormalization group methods in understanding critical phenomena and phase transitions. The review also highlights the connection between microphysics and macrophysics, the universality of critical behavior, and the importance of the renormalization group in quantum field theory and statistical physics.This review presents the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for quantum or thermal fluctuations. Non-perturbative solutions are obtained from approximations to the general form of the coarse-grained free energy or effective average action. These solutions interpolate between microphysical laws and complex macroscopic phenomena. The approach provides a unified description of O(N)-symmetric scalar models in two, three, or four dimensions, covering critical phenomena for second-order phase transitions, including the Kosterlitz-Thouless transition and polymer chain behavior. The critical equation of state is computed, establishing a connection between microphysical and critical quantities for liquid-gas transitions. Universal features of first-order phase transitions are studied in scalar matrix models. The quantitative treatment of coarse graining is essential for estimating nucleation rates. Quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons are discussed, as well as high-temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. The work explores chiral symmetry breaking and high-temperature or high-density chiral phase transitions in quantum chromodynamics using effective four-fermion interactions. The review discusses the non-perturbative flow equation, its solutions, and applications to scalar models, matrix models, and phase transitions. It emphasizes the importance of the effective average action in non-perturbative approximations and the role of renormalization group methods in understanding critical phenomena and phase transitions. The review also highlights the connection between microphysics and macrophysics, the universality of critical behavior, and the importance of the renormalization group in quantum field theory and statistical physics.