  We investigate the growth of the Nevanlinna Characteristic of f(z+\eta) for a
fixed \eta in this paper. In particular, we obtain a precise asymptotic
relation between T(r,f(z+\eta) and T(r,f), which is only true for finite order
meromorphic functions. We have also obtained the proximity function and
pointwise estimates of f(z+\eta)/f(z) which is a version of discrete analogue
of the logarithmic derivative of f(z). We apply these results to give growth
estimates of meromorphic solutions to higher order linear difference equations.
This also solves an old problem of Whittaker concerning a first order
difference equation. We show by giving a number of examples that all these
results are best possible in certain senses. Finally, we give a direct proof of
a result by Ablowitz, Halburd and Herbst.
