We describe a recent program whose main, general question could be phrased as follows: Given a structure $\mathcal M$ "made up of" two structures $\mathcal X_1$ and $\mathcal X_2$, can we "decompose" groups definable in $\mathcal M$ into groups, or sets, "coming from" $\mathcal X_1$ and $\mathcal X_2$? The purpose of this talk is to give precise meaning to the expressions in quotation marks in at least two different ways, report progress on the corresponding questions, and provide examples. The two main contexts will be when $\mathcal M$ is the disjoint union of $\mathcal X_1$ and $\mathcal X_2$, and when it is an expansion of $\mathcal X_1$ by a structure $\mathcal X_2$ whose universe is contained in that of $\mathcal X_1$.