This paper reviews quantum detection and estimation theory, which extends classical statistical decision and estimation theory to quantum systems. In classical statistics, probability density functions (p.d.f.) describe the likelihood of outcomes, while in quantum mechanics, density operators replace p.d.f.s. The paper discusses optimal procedures for distinguishing between two hypotheses, such as signal detection in noise, and quantum counterparts of classical estimation techniques like the Cramér-Rao inequality. It also covers quantum estimation theory, which seeks the best estimators of parameters of a density operator, and the lower bounds on mean-square errors of such estimates. The applications are primarily to the detection and estimation of optical frequency signals in thermal radiation. The paper also discusses quantum decision theory, which involves choosing between hypotheses based on quantum-mechanical density operators, and quantum estimation theory, which involves estimating parameters of a density operator. The paper concludes with a discussion of threshold detection, a method for detecting signals in noise at small signal-to-noise ratios, and the quantum threshold receiver, which is a quantum-mechanical counterpart of the classical threshold receiver. The paper also discusses the detection of coherent signals in thermal radiation and the detection of Gaussian radiation. The paper concludes with a discussion of the reception at an aperture, which is an important aspect of optical detection. The paper also discusses the choice among many hypotheses, which involves making decisions based on multiple hypotheses and the associated costs. The paper concludes with a discussion of the quantum threshold operator, which is used to detect signals in noise, and the moment-generating function of the threshold statistic, which is used to approximate the false-alarm and detection probabilities. The paper also discusses the application of quantum detection and estimation theory to the detection of incoherent light in the presence of thermal background radiation.This paper reviews quantum detection and estimation theory, which extends classical statistical decision and estimation theory to quantum systems. In classical statistics, probability density functions (p.d.f.) describe the likelihood of outcomes, while in quantum mechanics, density operators replace p.d.f.s. The paper discusses optimal procedures for distinguishing between two hypotheses, such as signal detection in noise, and quantum counterparts of classical estimation techniques like the Cramér-Rao inequality. It also covers quantum estimation theory, which seeks the best estimators of parameters of a density operator, and the lower bounds on mean-square errors of such estimates. The applications are primarily to the detection and estimation of optical frequency signals in thermal radiation. The paper also discusses quantum decision theory, which involves choosing between hypotheses based on quantum-mechanical density operators, and quantum estimation theory, which involves estimating parameters of a density operator. The paper concludes with a discussion of threshold detection, a method for detecting signals in noise at small signal-to-noise ratios, and the quantum threshold receiver, which is a quantum-mechanical counterpart of the classical threshold receiver. The paper also discusses the detection of coherent signals in thermal radiation and the detection of Gaussian radiation. The paper concludes with a discussion of the reception at an aperture, which is an important aspect of optical detection. The paper also discusses the choice among many hypotheses, which involves making decisions based on multiple hypotheses and the associated costs. The paper concludes with a discussion of the quantum threshold operator, which is used to detect signals in noise, and the moment-generating function of the threshold statistic, which is used to approximate the false-alarm and detection probabilities. The paper also discusses the application of quantum detection and estimation theory to the detection of incoherent light in the presence of thermal background radiation.