TITLE: Additive Subgroups of the Reals.
QUESTION [24 upvotes]: Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties of or interesting facts about these subgroups?

REPLY [4 votes]: Slightly off-topic, a weird subgroup in two dimensions is constructed in this paper: 
Ryuji Maehara. On a connected dense proper subgroup of ${\bf R}^2$ whose complement is connected . Proc. Amer. Math. Soc. 97 (1986) 556-558. MR 840645. 
That subgroup is constructed as a graph of a wild group homomorphism $f:\mathbf{R}\to \mathbf{R}$, so it is also a subgroup:
$\{ (x,f(x))  \mid x\in \mathbf{R}\}$