Number theoretic transforms (NTTs) in general and Fermat number transforms (FNTs) in particular have warranted considerable attention over the last two decades. The purpose of this research has been to investigate such transforms with the aim of enhancing their performance in digital signal and image processing and. overcoming problems that limit their use. Consequently, some new algorithms and ideas are introduced. With the objective of increasing the NTTs ability in carrying out convolutions and correlations, a new method is introduced, which extends the convolution length that can be calculated using the multiplication-free FNT beyond the previously well known limits. Also part of this thesis is devoted to the analysis of the NTTomain which is still unexploited Particular attention is given to periodics which are shown to have regular structures proving that the number domain is weIl defined. The frequency concept is related to the zero values which appear in the NTT domain of repetitive patterns, also some relevant theorems are derived. It is also shown that . NTTs are highly sensitive to small errors in the data, consequently, new applications are suggested. The discrete Hartley transform (DHT) has been proved to be a veryeffective transfom. In this thesis,. it is shown that the DHT can be mapped to circular convolution which can be efficiently calculated via the FNTs, so reducing dramatically the number of multiplications required at the expense of more easily executed shift and add operations. The method is also extended to higher dimensions. Another point which stems from this research is the revelation of an important connection between the NTTs and the Walsh-Hadamard transform (WHT). This has led to the derivation of a block -diagonal matrix with many zero values, which, unusually, allows the computation of the FNT from the WHT and vice versa, leading to a new method for calculating the FNT and the WHT or one from the one known. Finally, the variation in the efficiency of the transforms when dealing with different applications has raised the need for algorithms combining different transforms in order to achieve a better performance and a wider choice. A new fast and efficient algorithm is introduced in this thesis, which calculates both the FNT and the WHT or one from the other. It is also extended to the calculation of the discrete FoUrier transform (DFT) and compared with existing algorithms.