This Colloquium reviews quantum fluctuation relations and their applications. The key ingredients in the foundation of fluctuation relations are microreversibility and the Gibbs canonical ensemble. These relations are exact results that underpin nonequilibrium thermodynamics beyond the linear response regime. The material is organized to highlight the historical connection between quantum fluctuation relations and nonlinear response theory. The focus is on work fluctuation relations for transiently driven closed or open quantum systems and fluctuation relations for heat and matter exchange in quantum transport settings. Recent experimental applications are also discussed. The paper begins with an introduction to fluctuation relations, emphasizing their role in characterizing the statistical properties of fluctuating quantities. It then discusses nonlinear response theory and classical fluctuation relations, including the principle of microreversibility and the Bochkov-Kuzovlev approach. The paper addresses fundamental issues such as the distinction between inclusive and exclusive work, the problem of gauge freedom, and the fact that work is not a quantum observable. The paper then presents quantum work fluctuation relations, including the microreversibility of non-autonomous quantum systems, the work probability density function, and the characteristic function of work. It also discusses quantum exchange fluctuation relations and experimental applications, including proposals for experiments involving trapped cold ions and circuit quantum electrodynamics. The paper concludes with an outlook on future research directions.This Colloquium reviews quantum fluctuation relations and their applications. The key ingredients in the foundation of fluctuation relations are microreversibility and the Gibbs canonical ensemble. These relations are exact results that underpin nonequilibrium thermodynamics beyond the linear response regime. The material is organized to highlight the historical connection between quantum fluctuation relations and nonlinear response theory. The focus is on work fluctuation relations for transiently driven closed or open quantum systems and fluctuation relations for heat and matter exchange in quantum transport settings. Recent experimental applications are also discussed. The paper begins with an introduction to fluctuation relations, emphasizing their role in characterizing the statistical properties of fluctuating quantities. It then discusses nonlinear response theory and classical fluctuation relations, including the principle of microreversibility and the Bochkov-Kuzovlev approach. The paper addresses fundamental issues such as the distinction between inclusive and exclusive work, the problem of gauge freedom, and the fact that work is not a quantum observable. The paper then presents quantum work fluctuation relations, including the microreversibility of non-autonomous quantum systems, the work probability density function, and the characteristic function of work. It also discusses quantum exchange fluctuation relations and experimental applications, including proposals for experiments involving trapped cold ions and circuit quantum electrodynamics. The paper concludes with an outlook on future research directions.