It is a known fact that a recurrent point remains recurrent in pair with any other recurrent point (i.e. both points have synchronous returns to their arbitrarily neighborhoods) iff it is distal (it is so-called product recurrence). It is also know that system is (topologically) disjoint with any distal system iff it is minimal and weakly mixing.

Both proofs strongly use special recurrent properties of distal points. The above characterizations are also motivation for similar questions, e.g what are exactly the systems disjoint with all minimal systems, or what are points which are recurrent with all minimal points? Obviously, we can consider here different classes of recurrent points as well.

The aim of this talk is to survey through known results related to product recurrence and disjointness, presenting some known characterizations, remaining open problems and partial answers in the form of necessary or sufficient conditions that must/can be satisfied by a system in the considered class.