Recently Sklinos proved that every uncountable free group is not $\aleph_1$-homogenenous. This was later generalised by Belegradek to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this talk, we will present a joint work with G. Paolini that answers Belegradek's question in the negative by using some techniques inspired by the classical analysis of relatively free groups in infinitary logic. Our methods are general: they also apply to varieties with torsion and, more generally, to any variety of algebras in a countable language. For instance, we can prove that if V is a group variety containing a non-solvable group, then any uncountable V-free group is not $\aleph_1$-homogenenous, and if R is a countable ring, then any uncountable free left R-module is $\aleph_1$-homogeneous if and only if R is left perfect. After discussing some of the main applications of our results to the homogeneity problem, we will show how the same techniques can be used to study the relationship between two notions of strong substructure in free algebras: the syntactic notion of elementary substructure and the algebraic notion of V-free factor. Finally, we will give a structure theorem characterising the homogeneity of uncountable free algebras in a countable language in terms of an easily verifiable combinatorial condition. This theorem gives a new perspective on some famous results of Eklof, Mekler and Shelah on the topic of almost free algebras, showing that the problem of establishing the (first-order) homogeneity of uncountable free algebras corresponds exactly to the problem of determining the axiomatizability of the class of free algebras and the number of non-isomorphic models of their theory in some suitable infinitary logics.