Abstract: By applying Nevanlinna's value distribution theory of meromorphic functions, the following problems are investigated: the existence of finite order transcendental meromorphic solutions of several kinds nonlinear difference equations, and differential-difference equations of the form ~f^n(z)+M(z,f)=h(z), where n(\geq3) is an integer, h(z) is a given non-vanishing rational function, and M(z,f) is a linear differential-difference polynomial with small meromorphic coefficients.