  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.
