This colloquium focuses on quantum fluctuation relations, which are fundamental in the development of nonequilibrium thermodynamics beyond the linear response regime. The authors provide a comprehensive overview of the theory and applications of these relations, emphasizing their historical connection to nonlinear response theory. The main topics covered include: 1. **Nonlinear Response Theory and Classical Fluctuation Relations**: - **Microreversibility of non-autonomous classical systems**: The principle of microreversibility is discussed for classical systems, including the time-reversed dynamics and the relationship between forward and backward processes. - **Bochkov-Kuzovlev approach**: This approach provides a compact classical expression for fluctuation relations, encompassing linear and nonlinear response theory, as well as the second law of thermodynamics. - **Jarzynski approach**: This approach defines inclusive work and introduces the Jarzynski equality, which relates the average work to the free energy difference between equilibrium states. 2. **Fundamental Issues**: - **Inclusive, exclusive, and dissipated work**: The definitions of inclusive and exclusive work are clarified, and the distinction between them is important for experimental applications. - **The problem of gauge freedom**: The gauge freedom in the definition of work is discussed, highlighting that different experimental setups imply different gauges. - **Work is not a quantum observable**: The quantum version of work is defined, emphasizing that it cannot be represented by a Hermitian operator and must be measured twice at different times. 3. **Quantum Work Fluctuation Relations**: - **Microreversibility of non-autonomous quantum systems**: The principle of microreversibility is extended to non-autonomous quantum systems, providing a foundation for quantum fluctuation relations. - **Derivation of quantum fluctuation relations**: The quantum analogs of classical fluctuation relations are derived, including the Bochkov-Kuzovlev and Jarzynski identities. 4. **Experiments**: - **Work fluctuation relations**: Experimental proposals using trapped cold ions and circuit quantum electrodynamics are discussed. - **Exchange fluctuation relations**: An electron counting statistics experiment and a nonlinear response experiment in a quantum coherent conductor are presented. The colloquium aims to clarify fundamental issues and provide a self-contained exposition of quantum fluctuation relations, emphasizing their historical development and recent advancements.This colloquium focuses on quantum fluctuation relations, which are fundamental in the development of nonequilibrium thermodynamics beyond the linear response regime. The authors provide a comprehensive overview of the theory and applications of these relations, emphasizing their historical connection to nonlinear response theory. The main topics covered include: 1. **Nonlinear Response Theory and Classical Fluctuation Relations**: - **Microreversibility of non-autonomous classical systems**: The principle of microreversibility is discussed for classical systems, including the time-reversed dynamics and the relationship between forward and backward processes. - **Bochkov-Kuzovlev approach**: This approach provides a compact classical expression for fluctuation relations, encompassing linear and nonlinear response theory, as well as the second law of thermodynamics. - **Jarzynski approach**: This approach defines inclusive work and introduces the Jarzynski equality, which relates the average work to the free energy difference between equilibrium states. 2. **Fundamental Issues**: - **Inclusive, exclusive, and dissipated work**: The definitions of inclusive and exclusive work are clarified, and the distinction between them is important for experimental applications. - **The problem of gauge freedom**: The gauge freedom in the definition of work is discussed, highlighting that different experimental setups imply different gauges. - **Work is not a quantum observable**: The quantum version of work is defined, emphasizing that it cannot be represented by a Hermitian operator and must be measured twice at different times. 3. **Quantum Work Fluctuation Relations**: - **Microreversibility of non-autonomous quantum systems**: The principle of microreversibility is extended to non-autonomous quantum systems, providing a foundation for quantum fluctuation relations. - **Derivation of quantum fluctuation relations**: The quantum analogs of classical fluctuation relations are derived, including the Bochkov-Kuzovlev and Jarzynski identities. 4. **Experiments**: - **Work fluctuation relations**: Experimental proposals using trapped cold ions and circuit quantum electrodynamics are discussed. - **Exchange fluctuation relations**: An electron counting statistics experiment and a nonlinear response experiment in a quantum coherent conductor are presented. The colloquium aims to clarify fundamental issues and provide a self-contained exposition of quantum fluctuation relations, emphasizing their historical development and recent advancements.