  New and old results on closed polynomials, i.e., such polynomials f in
K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n],
are collected. Using some properties of closed polynomials we prove the
following factorization theorem: Let f be an element of K[x_1,...,x_n], where K
is an algebraically closed field. Then for all but finite number of a's the
polynomial f+a can be decomposed into a product of irreducible polynomials of
the same degree (not depending on a) such that their defferences are constants.
An algorithm for finding of a generative polynomial of a given polynomial f,
which is a closed polynomial h with f=F(h) for some F(t) in K[t], is given.
Some types of saturated subalgebras A in K[x_1,...,x_n] are considered, i.e.,
such that for any f in A a generative polynomial of f is contained in A.
