The search for universal models began in the early twentieth century when Hausdorff showed that there is a universal linear order of cardinality $\aleph_{n+1}$ if $2^{\aleph_n}= \aleph_{n + 1}$, i.e., a linear order $U$ of cardinality $\aleph_{n+1}$ such that every linear order of cardinality $\aleph_{n+1}$ embeds in $U$. In this talk, we will study universal models in several classes of abelian groups and modules with respect to pure embeddings. In particular, we will present a complete solution below $\aleph_\omega$, with the exception of $\aleph_0$ and $\aleph_1$, to Problem 5.1 in page 181 of Abelian Groups by L\'{a}szl\'{o} Fuchs, which asks to find the cardinals $\lambda$ such that there is a universal abelian p-group for purity of cardinality $\lambda$. The solution presented will use both model-theoretic and set-theoretic ideas.