  These are the notes of my lectures at the 1996 European Congress of
Mathematicians. {} Polynomials appear in mathematics frequently, and we all
know from experience that low degree polynomials are easier to deal with than
high degree ones. It is, however, not clear that there is a well defined class
of "low degree" polynomials. For many questions, polynomials behave well if
their degree is low enough, but the precise bound on the degree depends on the
concrete problem. {} It turns out that there is a collection of basic questions
in arithmetic, algebraic geometry and topology all of which give the same class
of "low degree" polynomials. The aim of this lecture is to explain these
properties and to provide a survey of the known results.
