In this talk I will present ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word $\lambda$ and the prefixes of a right-infinite word $\rho$. I will talk about a result that states that the set of minimum lengths of words $x$ and $x'$ such that $x\lambda_n = \rho_nx'$ or $\lambda_nx = x'\rho_n$ is finite, where n runs over positive integers and $\lambda_n$ and $\rho_n$ are respectively the suffix of $\lambda$ and the prefix of $\rho$ of length $n$, if and only if $\lambda$ and $\rho$ are ultimately periodic words of the form $\lambda = u^{-\infty}v$ and $\rho = wu^{\infty}$ for some finite words $u$, $v$ and $w$. This is a joint work with José Carlos Costa (Universidade do Minho) and M. Lurdes Teixeira (Universidade do Minho).