  We determine all posible orders of automorphisms of finite order of complex
K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer
$N$ is the order of an automorphism of a complex K3 surface if and only if
$\phi(N)\le 20$ where $\phi$ is the Euler function. In particular, 66 is the
maximum finite order in each characteristic $p\neq 2,3$. As a consequence, we
give a bound for the orders of finite groups acting on K3 surfaces in
characteristic $p>7$.
