Piecewise hyperdefinable sets are natural generalizations of interpretable sets. A standard example is the quotient of a subgroup generated by a definable set over a type-definable normal subgroup. On the other hand, approximate subgroups are subsets of a group similar to subgroups up to a finite discrete-like error. Rough approximate subgroups generalise approximate subgroups by allowing also a no-discrete-like error. The most relevant case of rough approximate subgroups occurs in metric groups when the no-discrete error is given by the metric. Firstly, we will discuss the general structure of piecewise hyperdefinable groups. Then, we will see an application to rough approximate subgroups and some combinatorial consequences in the particular case of metric groups. All this corresponds to my Ph.D. thesis which is divided into two papers: \emph{On piecewise hyperdefinable groups} (arXiv:2011.11669) and \emph{On metric approximate subgroups} (joint with Hrushovski, soon in arxiv).