Abstract: Necklace polynomials arise in different areas of mathematics: combinatorics, arithmetic and Lie theory. In my talk, I will discuss their significance in each of these areas with special emphasis on a generalised Witt formula that one may deduce for graded Lie algebras. This formula can be used to prove a Gromov-like theorem for graded Lie algebras of type FP. Necklaces, finite fields, and Lie algebras