  In the paper, properties of orbit functions are reviewed and further
developed. Orbit functions on the Euclidean space $E_n$ are symmetrized
exponential functions. The symmetrization is fulfilled by a Weyl group
corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be
described. An orbit function is the contribution to an irreducible character of
a compact semisimple Lie group $G$ of rank $n$ from one of its Weyl group
orbits. It is shown that values of orbit functions are repeated on copies of
the fundamental domain $F$ of the affine Weyl group (determined by the initial
Weyl group) in the entire Euclidean space $E_n$. Orbit functions are solutions
of the corresponding Laplace equation in $E_n$, satisfying the Neumann
condition on the boundary of $F$. Orbit functions determine a symmetrized
Fourier transform and a transform on a finite set of points.
