We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mn\"{e}v and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which $\exists \mathbb{R}$ rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. $\exists \mathbb{R}$ has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for $\exists \mathbb{R}$, as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.

翻译：本文综述了复杂性类 $\exists \mathbb{R}$，该类刻画了判定实数量词存在理论的复杂性。$\exists \mathbb{R}$ 类源于两个不同的传统：一个基于 Blum-Shub-Smale 实数计算模型，另一个则遵循 Mnëv 和 Shor 关于有向拟阵实现空间普适性的研究。多年来，以 $\exists \mathbb{R}$（而非 NP）作为衡量其复杂性的恰当方式的问题数量不断增长，特别是在计算几何、图形绘制、博弈论以及逻辑与代数的某些领域。$\exists \mathbb{R}$ 也开始出现在机器学习、马尔可夫决策过程和概率推理的背景下。我们致力于编纂一部关于 $\exists \mathbb{R}$ 完全性及困难性问题的综合纲要，并附有大量待解问题列表。该纲要构成我们综述的第三部分；第二部分则是对纲要及其所涉领域的导览。第一部分向读者介绍了作为复杂性类的实数量词存在理论，讨论了其历史背景、研究动机、发展前景以及若干技术细节。