A tutorial-review paper on discrete orthogonal transforms and their applications in digital signal and image (both monochrome and color) processing is presented. Various transforms such as discrete Fourier, discrete cosine, Walsh-Hadamard, slant, Haar, discrete linear basis, Hadamard-Haar, rapid, lower triangular, generalized Haar, slant Haar and Karhunen-Loeve are defined and developed. Pertinent properties of these transforms such as power spectra, cyclic and dyadic convolution and correlation are outlined. Efficient algorithms for fast implementation of these transforms based on matrix partitioning or matrix factoring are presented. The application of these transforms in speech and image processing, spectral analysis, digital filtering (linear, nonlinear, optimal and suboptimal), nonlinear systems analysis, spectrography, digital holography, industrial testing, spectrometric imaging, feature selection, and patter recognition is presented. The utility and effectiveness of these transforms are evaluated in terms of some standard performance criteria such as computational complexity, variance distribution, mean-square error, correlated rms error, rate distortion, data compression, classification error, and digital hardware realization.