  We study properties of a subclass of Markov processes that have all moments
that are continuous functions of the time parameter and more importantly are
characterized by the property that say their $n-$th conditional moment given
the past is also a polynomial of degree not exceeding $n.$ Of course all
processes with independent increments with all moments belong to this class. We
give characterization of them within the studied class. We indicate other
examples of such process. Besides we indicate families of polynomials that have
the property of constituting martingales. We also study conditions under which
processes from the analysed class have orthogonal polynomial martingales and
further are harnesses or quadratic harnesses. We provide examples illustrating
developed theory and also provide some interesting open questions. To make
paper interesting for a wider range of readers we provide short introduction
formulated in the language of measures on the plane.
