A fundamental area of analysis concerns the classification of dynamical systems, both topological and measure theoretic in nature. In particular, if a dynamical system is both measure preserving and topological in nature, the most natural question which arises is which finitary isomorphism class this system belongs to. Already in 1979, for example, I have shown in an article in the Annals of Mathematics that two Bernoulli schemes of the same entropy are finitarily isomorphic; this article initiated much research which is still going on in this area. Recently I have discovered that also orbit equivalence is amenable to finitary classification, which yields simpler proofs than those known before, for example, for Dye's theorem. The main question is to determine whether any two measure preserving transformations consisting of shift spaces with finite alphabets together with finite invariant ergodic measures are finitarily orbit equivalent. This has now been shown for monothetic rotations of compact abelian groups. The project consists in showing that this theorem is also valid for many other such transformations, including Bernoulli schemes, and in finding an example, yet unknown, in which this theorem fails. It is still possible that the theorem is valid in general for shift spaces. This question also arises in number theoretic transformations, such as the natural extension of the continued fraction transformation, for which the answer is yet unknown. Answers to these and related questions will be very stimulating and instructive for future research in ergodic theory and dynamical systems.