This is the first part of a two-paper series, in which we critically examine the various proposals that have been made for superluminal coordinate transformations. Here we consider the two-dimensional case. Starting from rather general assumptions, we show that the superluminal coordinate transformations in two dimensions are essentially uniquely determined. Different proposals for such transformations are then analyzed from the point of view of those assumptions. The relationship between the superluminal transformations and the discrete symmetries P (parity), T (time reversal), and PT is also discussed.