The classification of non-singular dynamical systems up to orbit equivalence received a lot of attention in the latter part of the last century. The Dye-Krieger Theorem says that each such system can be realised as an odometer acting on a product space, but says nothing about the measure which might arise when this equivalence is implemented. By the Connes-Krieger classification, there is only one class of systems in the case of measure-preserving systems or those of type $III_{\lambda}$, for $0 <{\lambda} \leq 1$. However, in the case of systems of type $III_0$, Krieger gave an example of a dynamical system which is not orbit equivalent to one with a product measure. (They are, of course, classified up to equivalence by their associated flows, but these are difficult to compute.)
When the system is measure-preserving, the Jewett-Krieger theorem says that every ergodic system is orbit equivalent to one which is uniquely ergodic, with a nice kind of orbit equivalence map.
In joint work with Toshi Hamachi, we have recently shown that every ergodic non-singular system is orbit equivalent to one where the measure is Markov: these two-state measures are only one step more complicated than product measures. Considering Markov measures as a special case of $G$-measures, there is a natural notion of unique ergodicity; we can show that the system may be chosen to have this property, an analogue of Jewett-Krieger.
We also have some very explicit examples of non-product Markov systems.
This represents a small step towards the ultimate goal of classifying type $III_0$ systems. I will aim to give an overview of this fascinating subject.