In a block adaptive filtering procedure, the filter coefficients are adjusted once per each output block while maintaining performance comparable to that of widely used Least Mean Square (LMS) adaptive filtering in which the filter coefficients are adjusted once per each output data sample. In this block adaptive filtering procedure, the output values and gradient estimate are given as convolution and correlation, respectively. Accordingly, an efficient implementation of block adaptive filters is possible by means of discrete transform technique which has cyclic convolution property and fast algorithms. In this thesis, the block adaptive filtering using Fermat Number Transform (FNT) is studied to exploit the computational efficiency and less quantization effect on the performance compared with finite precision FFT realization. It is shown that the block adaptive filters using FNT have advantages over those using FFT in terms of computational complexity and adaptation accuracy. And this has been verified by computer simulation for several applications including adaptive channel equalizer (PAM and QAM), adaptive line enhancer, and system identification. In addition, we have formulated a two dimensional block adaptive filtering procedure in which the number theoretic transform such as FNT can be used efficiently.