In this paper, the concept of the two-dimensional discrete Fourier transformation (2-D DFT) is defined in the general case, when the form of relation between the spatial-points (x, y) and frequency-points (ω1, ω2) is defined in the exponential kernel of the transformation by a nonlinear form L(x, y; ω1, ω2). The traditional concept of the 2-D DFT uses the Diaphanous form xω1 +yω2 and this 2-D DFT is the particular case of the Fourier transform described by the form L(x, y; ω1, ω2). Properties of the general 2-D discrete Fourier transform are described and examples are given. The special case of the N × N-point 2-D Fourier transforms, when N = 2r, r > 1, is analyzed and effective representation of these transforms is proposed. The proposed concept of nonlinear forms can be also applied for other transformations such as Hartley, Hadamard, and cosine transformations.