  We study general properties for the family of stochastic processes with
polynomial regression property, that is that every conditional moment of the
process is a polynomial. It turns out that then there exists a family of
polynomial martingales $\left\{ M_{n}(X_{t},t)\right\}_{n\geq1}$ that contains
complete information on the distribution (both marginal and transitional) of
the process.
  We specify conditions expressed in terms of $M_{n}^{\prime}s$ under which a
given process has independent increments and further is a Levy process,
contains reversed martingales, is a harness or quadratic harness. We also give
conditions under which some of these martingales are also reversed martingales.
