I am currently learning the theory of Shimura varieties. Out of curiosity, is it known which number fields can occur as reflex fields? More precisely, can one find, for any number field, a positive dimensional Shimura variety which has this field as its reflex field?



Reflex fields of Shimura varieties
The answer depends on your definition of a Shimura pair $(G,X)$.


Look in Section 2.1 of [Deligne's paper](https://publications.ias.edu/sites/default/files/34_VarietesdeShimura.pdf).
If you assume only axioms (2.1.1.1), (2.1.1.2) and (2.1.1.3), then any number field $F$ can occur as the reflex field $E(G,X)$ with $X$ of positive dimension. 
Indeed, take $G\_1=R\_{F/\mathbf{Q}}\mathbb{G}\_{m,F}$, then for suitable $h\_1\colon\mathbf{S}\to G\_{1,\mathbf{R}}$ we have $E(G\_1,h\_1)=F$. 
We take $X\_1=\{h\_1\}$, then $\mathrm{dim}\_\mathbf{C}(X\_1)=0$. We take $(G\_2,X\_2)=(\mathrm{GL}\_{2,\mathbf{Q}},X\_2)$ (the standard Shimura pair for $\mathrm{GL}\_{2}$), then $E(G\_2,X\_2)=\mathbf{Q}$, $\mathrm{dim}(X\_2)=1$. 
Set $G=G\_1\times\_\mathbf{Q} G\_2$, $X=X\_1\times X\_2$. 
Then $\mathrm{dim}\_\mathbf{C}(X)=1$, $E(G,X)=E(G\_1,X\_1)=F$.


However, if you assume also axioms (2.1.1.4) and (2.1.1.5), then $E(G,X)$ must be either a totally real field or a CM-field. 
In order to show this, it suffices to consider the case when $G$ is $\mathbf{Q}$-simple adjoint and the case when $G$ is a torus. 
For the $\mathbf{Q}$-simple adjoint case see Section 2.3.4 and Proposition 2.3.6 in Deligne's paper. In the toric case, axioms (2.1.1.4) and (2.1.1.5) imply that the torus $G$ is isogenous to the product of $\mathbb{G}\_{m,\mathbf{Q}}$ and a $\mathbf{Q}$-torus which is compact over $\mathbf{R}$, hence it splits over a CM-field, and the assertion follows.


