The real part of ${\varepsilon }^{\prime }/\varepsilon$ measures direct $\mathrm{CP}$ violation in the decays of neutral kaons into two pions. This is a fundamental quantity that has justly attracted a great deal of theoretical as well as experimental work. Determination of its value may answer the question of whether $\mathrm{CP}$ violation is present only in the mass matrix of neutral kaons (the superweak scenario) or whether it is also at work directly in the decay amplitudes. After a brief historical summary, the present and expected experimental sensitivities are discussed. In light of these, the authors address the problem of estimating ${\varepsilon }^{\prime }/\varepsilon$ in the standard model and review the status of the theoretical predictions of ${\varepsilon }^{\prime }/\varepsilon$ as of the beginning of 1999. The short-distance part of the computation is now known to the next-to-leading order in QCD and QED and is therefore well under control. On the other hand, the evaluation of the hadronic matrix elements of the relevant operators is where most of the theoretical uncertainty still resides. The authors analyze the results of the most extensive calculations to date. The values of the matrix-element parameters in the various approaches are discussed, together with the allowed range of quark mixing angles in the Cabibbo-Kobayashi-Maskawa matrix. All recent predictions of ${\varepsilon }^{\prime }/\varepsilon$ are summarized and compared. Because of the intrinsic uncertainties of the long-distance computations, values ranging from ${10}^{-4}$ to a few times ${10}^{-3}$ can be accounted for in the standard model. Since this range covers most of the present experimental uncertainty, it is unlikely that new-physics effects can be disentangled from the standard-model prediction.

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