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We study limit models in the class of abelian groups with the subgrouprelation and in the class of torsion-free abelian groups with the pure subgrouprelation. We show:  $\textbf{Theorem}$  (1) If $G$ is a limit model of cardinality $\lambda$ in the class of abeliangroups with the subgroup relation, then $G \cong (\oplus_{\lambda}\mathbb{Q})\oplus \oplus_{p \text{ prime}} (\oplus_{\lambda} \mathbb{Z}(p^\infty))$.  (2) If $G$ is a limit model of cardinality $\lambda$ in the class oftorsion-free abelian groups with the pure subgroup relation, then:  * If the length of the chain has uncountable cofinality, then $ G \cong(\oplus_{\lambda} \mathbb{Q} ) \oplus \Pi_{p\text{ prime}}\overline{(\oplus_{\lambda} \mathbb{Z}_{(p)})}$.  * If the length of the chain has countable cofinality, then $G$ is notalgebraically compact.  We also study the class of finitely Butler groups with the pure subgrouprelation, we show that it is an AEC, Galois-stable and $(<\aleph_0)$-tame andshort.

Algebraic description of limit models in classes of abelian groups