The unary polynomial functions on any algebra $\ab{A}$ are those functions
that can be obtained from the identity function and the
constant functions by using the fundamental operations
of $\mathbf{A}$. Given an algebra $\mathbf{A}$, we try to determine
which functions are polynomial. Only recently, P. Mayr and
the speaker have determined the number of polynomial functions
of certain linear groups.
Often, one attempts to describe polynomial functions
by preservation properties; for example, every
polynomial function preserves congruences. For some
algebras, this property actually characterizes polynomial functions.
We will exhibit some results (by P. M. Idziak and the speaker) that describe
those finite commutative rings with $1$ on which every
congruence preserving function is polynomial.