Expanding a model theoretically ?tame? structure in a way that it stays ?tame? has been a theme in the recent years. In the first part of this talk, we present a history of work done in that frame. Then we focus on the case of expansions of o-minimal structures by a unary predicate. There is a dividing line according to whether the predicate is dense or discrete; even though the results obtained are similar, there is an enormous difference in the techniques used. We shall present some of the results obtained in the dense case. Starting from a set of abstract axioms, we obtain a decomposition theorem for definable sets and a local structure theorem for definable groups. The abstract axioms mentioned above are ?smallness?, ?o-minimal open core? and ?quantifier elimination up to existential formulas?. We shall illustrate a proof of the fact that the first two imply ?quantifier elimination up to bounded formulas?, which is a weak form of the last axiom and we give reasons why it is really weaker than that axiom. (Joint work with P. Eleftheriou and P. Hieronymi)