  Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics.
Recently, several refinements of this seminal result were obtained for special,
more structured classes of graphs. We survey these results in their rich
combinatorial context. In particular, we stress the link to the theory of
(structural) sparsity, which leads to alternative proofs, refinements and
solutions of open problems. It is interesting to note that many of these
classes present challenging problems. Nevertheless, from the point of view of
regularity lemma type statements, they appear as "gentle" classes.
