  The effects of finite size and of finite current on the current-voltage
characteristics of Josephson junction arrays is studied both theoretically and
by numerical simulations. The cross-over from non-linear to linear behavior at
low temperature is shown to be a finite size effect and the non-linear behavior
at higher temperature, $T>T_{KT}$, is shown to be a finite current effect.
These are argued to result from competition between the three length scales
characterizing the system. The importance of boundary effects is discussed and
it is shown that these may dominate the behavior in small arrays.
