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Many scientific papers treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This paper presents a new description of three essential classes of fractal dimensions based on: (i) morphology, (ii) entropy, and (iii) transforms, all unified through the generalized entropy-based Renyi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this paper is unique in terms of (i) its progressive development of the fractal dimension concept, (ii) similarity in the definitions and expressions, (iii) analysis of the relation between the dimensions, and (iv) their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioural processes such as dishabituation, irreversible and birth-death growth phenomena e.g., diffusion-limited aggregates, DLAs, dielectric discharges, and cellular automata, as well as dynamical nonstationary transient processes such as speech and transients in radio transmitters, multifractal optimization of image compression using learned vector quantization with Kohonen 's self-organizing feature maps (SOFMs), and multifractal-based signal denoising.