The paper reviews the Kirkwood–Dirac (KD) distribution, a quasi-probability distribution that has gained prominence in recent years as a powerful tool for analyzing quantum mechanics. Unlike the Wigner function, which is well-suited for continuous-variable systems but not for finite-dimensional systems and general observables, the KD distribution allows for the representation of quantum states using arbitrary observables. The review is divided into three parts: definitions and basic properties, applications in various areas of quantum mechanics, and mathematical structure. 1. **Definitions and Basic Properties**: The KD distribution is introduced, focusing on its standard form and generalizations. It is defined in terms of two orthonormal bases in a complex Hilbert space, and its components are expressed as inner products of states and observables. The distribution satisfies some but not all axioms of joint probability distributions, allowing for negative or non-real values, which are crucial for describing non-classical phenomena. 2. **Applications in Quantum Mechanics**: - **Quantum Metrology**: Non-real entries in KD distributions play a key role in enhancing measurement precision. The imaginary part of KD quasi-probabilities encodes disturbance in quantum states, and non-real components can break classical bounds on metrological information distillation. - **Weak Values**: Weak values, which are averages of observables conditioned on pre- and post-selected states, are closely related to the KD distribution. They can be measured using weak-value amplification techniques, which improve the signal-to-noise ratio in metrological estimation. - **Other Applications**: The KD distribution is also used in quantum thermodynamics, quantum chaos, and foundational studies of quantum mechanics, including Leggett–Garg inequalities and contextuality. 3. **Mathematical Structure**: The paper discusses the geometry of KD-positive states, methods to witness and quantify KD non-positivity, and relationships between KD non-positivity and observables' incompatibility. It highlights the importance of understanding the mathematical properties of the KD distribution for its effective application in quantum information processing. The review emphasizes the operational advantages of using the KD distribution in quantum mechanics, particularly in scenarios where classical probability distributions are insufficient.The paper reviews the Kirkwood–Dirac (KD) distribution, a quasi-probability distribution that has gained prominence in recent years as a powerful tool for analyzing quantum mechanics. Unlike the Wigner function, which is well-suited for continuous-variable systems but not for finite-dimensional systems and general observables, the KD distribution allows for the representation of quantum states using arbitrary observables. The review is divided into three parts: definitions and basic properties, applications in various areas of quantum mechanics, and mathematical structure. 1. **Definitions and Basic Properties**: The KD distribution is introduced, focusing on its standard form and generalizations. It is defined in terms of two orthonormal bases in a complex Hilbert space, and its components are expressed as inner products of states and observables. The distribution satisfies some but not all axioms of joint probability distributions, allowing for negative or non-real values, which are crucial for describing non-classical phenomena. 2. **Applications in Quantum Mechanics**: - **Quantum Metrology**: Non-real entries in KD distributions play a key role in enhancing measurement precision. The imaginary part of KD quasi-probabilities encodes disturbance in quantum states, and non-real components can break classical bounds on metrological information distillation. - **Weak Values**: Weak values, which are averages of observables conditioned on pre- and post-selected states, are closely related to the KD distribution. They can be measured using weak-value amplification techniques, which improve the signal-to-noise ratio in metrological estimation. - **Other Applications**: The KD distribution is also used in quantum thermodynamics, quantum chaos, and foundational studies of quantum mechanics, including Leggett–Garg inequalities and contextuality. 3. **Mathematical Structure**: The paper discusses the geometry of KD-positive states, methods to witness and quantify KD non-positivity, and relationships between KD non-positivity and observables' incompatibility. It highlights the importance of understanding the mathematical properties of the KD distribution for its effective application in quantum information processing. The review emphasizes the operational advantages of using the KD distribution in quantum mechanics, particularly in scenarios where classical probability distributions are insufficient.