  In this paper we study multiple ergodic averages for "good" variable
polynomials. In particular, under an additional assumption, we show that these
averages converge to the expected limit, making progress related to an open
problem posted by Frantzikinakis (Problem 10, "Some open problems on multiple
ergodic averages. Bulletin of the Hellenic Mathematical Society. 60 (2016),
41-90"). Corresponding averages along prime numbers are studied too. These
general convergence results imply various variable extensions of classical
recurrence, combinatorial and number theoretical results which are presented as
well.
