Conjugacy of real diffeomorphisms. A survey

Given a group $G$, the conjugacy problem in $G$ is the problem of giving an effective procedure for determining whether or not two given elements $f,g\in G$ are conjugate, i.e. whether there exists $h\in G$ with $fh=hg$. This paper is about the conjugacy problem in the group $\mathrm{Diffeo}(I)$ of all diffeomorphisms of an interval $I\subset\mathbb R$.
There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.