In this thesis, we present a theoretical study of biorthogonal Coifman wavelet systems, a family of biorthogonal wavelet systems with vanishing moments equally distributed between scaling functions and wavelet functions. One key property of these wavelet systems is that they provide nice wavelet sampling approximation with exponential decay. Moreover they are compactly supported, symmetric, have growing smoothness with large degrees, and converge to the sinc wavelet system. Using a time domain design method, the exact formulas of the coefficients of biorthogonal Coifman wavelet systems of all degrees are obtained. An attractive feature behind it is that all the coefficients are dyadic rational, which means that we can implement a very fast multiplication-free discrete wavelet transform, which consists of only addition and shift operations, on digital computers. The transform coding performance of biorthogonal Coifman wavelet systems is quite comparable to other widely used wavelet systems. The orthogonal counterparts, orthogonal Coifman wavelet systems, are also discussed in this thesis. In addition we develop a new wavelet-based embedded image coding algorithm, the Wavelet-Difference-Reduction algorithm. Unlike zerotree type schemes which use spatial orientation tree structures to implicitly locate the significant wavelet transform coefficients, this new algorithm is a direct approach to find the positions of significant coefficients. It combines the discrete wavelet transform, differential coding, binary reduction, ordered bit plane transmission, and adaptive arithmetic coding. The encoding can be stopped at any point, which allows a target rate or distortion metric to be met exactly; the decoder can also terminate the decoding at any point, and produce a corresponding reconstruction image. Our algorithm provides a fully embedded code to successively approximate the original image source; thus it's well suited for progressive image transmission. It is very simple in its form (which will make the encoding and decoding very fast), and has a clear geometric structure, which enables us to process the image data in the compressed wavelet domain. The image coding results of it are quite competitive with almost all previous reported image compression algorithms on standard test images.