The problem of designing wavelets which are most appropriate for applications to multiresolution coding of image, speech, radar and other signals is addressed. The effects of regularity and zero moments on the design of wavelets and filter banks used to realize these wavelet decompositions are discussed, and insights pointed out. The use of vector quantization with wavelet transforms will be discussed. It is observed how wavelet decompositions are a compromise between optimality and complexity, where the optimality is determined from the minimization of bit rate and distortion, using rate distortion theory. The problem of designing wavelets yielding linear phase filtering, important for applications such as television coding and radar, is discussed and a number of approaches to solutions are described. These include the use of biorthogonal rather than orthogonal bases for wavelets which are realizable by general perfect reconstruction filter banks in which the analysis and synthesis filters are not time-reversed versions of each other. Methods for designing linear phase filters are briefly discussed and referenced. In the discussion on applications to radar signals, the relation of wavelet theory to a special signal called a chirplet is noted. Some connections of wavelets to splines and cardinal series are noted. Finally, wavelets which almost meet the uncertainty principle bound with equality are described.