The Kirkwood–Dirac (KD) distribution is a quasi-probability distribution that allows the application of statistical and probability theory to quantum mechanics. Unlike the Wigner function, which is limited to continuous-variable systems, the KD distribution can represent quantum states in terms of arbitrary observables, making it suitable for finite-dimensional systems and general observables. This paper reviews the KD distribution, its properties, and its applications in quantum mechanics. The KD distribution is defined in terms of a matrix representation of a quantum state with respect to two orthonormal bases. It satisfies some, but not all, of Kolmogorov's axioms for joint probability distributions and can have negative or non-real values, which are considered non-classical. The KD distribution has been used in various areas of quantum mechanics, including quantum metrology, weak values, direct measurements of quantum states, quantum thermodynamics, and the foundations of quantum mechanics. In quantum metrology, non-real KD quasi-probabilities are essential for accessing unknown information encoded in quantum states. The KD distribution also plays a role in post-selected quantum metrology, where it enables the distillation of quantum information from many particles into a few. The KD distribution's mathematical structure has been studied, and its properties, such as non-positivity, have been quantified. The paper also discusses the connection between KD non-positivity and quantum contextuality, as well as the use of the KD distribution in weak-value amplification experiments. Overall, the KD distribution provides a powerful tool for analyzing quantum mechanics and has become an important mathematical framework in quantum theory.The Kirkwood–Dirac (KD) distribution is a quasi-probability distribution that allows the application of statistical and probability theory to quantum mechanics. Unlike the Wigner function, which is limited to continuous-variable systems, the KD distribution can represent quantum states in terms of arbitrary observables, making it suitable for finite-dimensional systems and general observables. This paper reviews the KD distribution, its properties, and its applications in quantum mechanics. The KD distribution is defined in terms of a matrix representation of a quantum state with respect to two orthonormal bases. It satisfies some, but not all, of Kolmogorov's axioms for joint probability distributions and can have negative or non-real values, which are considered non-classical. The KD distribution has been used in various areas of quantum mechanics, including quantum metrology, weak values, direct measurements of quantum states, quantum thermodynamics, and the foundations of quantum mechanics. In quantum metrology, non-real KD quasi-probabilities are essential for accessing unknown information encoded in quantum states. The KD distribution also plays a role in post-selected quantum metrology, where it enables the distillation of quantum information from many particles into a few. The KD distribution's mathematical structure has been studied, and its properties, such as non-positivity, have been quantified. The paper also discusses the connection between KD non-positivity and quantum contextuality, as well as the use of the KD distribution in weak-value amplification experiments. Overall, the KD distribution provides a powerful tool for analyzing quantum mechanics and has become an important mathematical framework in quantum theory.