  We investigate the existence of non-trivial holomorphic and meromorphic
solutions of Fermat functional equations over an open Riemann surface $S$. When
$S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always
exists non-trivial holomorphic and meromorphic solution. When $S$ is a general
open Riemann surface, we prove that every non-trivial holomorphic or
meromorphic solution satisfies a growth condition, provided that the power
exponents of the equations are bigger than some certain positive integers.
