Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in R$^n$. A (not necessarily normal) affine toric variety is given by finite subset A of Z$^n$. When A is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R$^n$. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A.

Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R$^n$. These are R$^n$$_>$-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R$^n$ is homeomorphic to the secondary polytope of A.

This talk will sketch this story of irrational toric varieties. It represents work with Garcia-Puente, Zhu, Postinghel, and Villamizar, and work in progress with Pir.