Compression of vector-valued datasets is increasingly needed for addressing the significant storage and transmission burdens associated with research activities in large-scale computational fluid dynamics and environmental science. However, vector-valued compression schemes have traditionally received few investigations within the data-compression community. Consequently, this dissertation conducts a systematic study of effective algorithms for the coding of vectorvalued datasets and builds practical embedded compression systems for both static and timevarying vector fields. In generalizing techniques from the relatively mature field of image and video coding to vector data, three critical issues must be addressed: the design of a vector wavelet transform (VWT) that is amenable to vector-valued compression applications, the implementation of vector-valued intraframe coding that enables embedded coding, and the investigation of interframe-compression techniques that are appropriate for the complex temporal evolutions of vector features. In this dissertation, we initially invoke multiwavelets to construct VWTs. However, a balancing problem arises when existing multiwavelets are applied directly to vector data. We analyze extensively this performance failure and develop the omnidirectional balancing (OB) design criterion to rectify it. Employing the OB principle, we derive with a family of biorthogonal multiwavelets possessing desired balancing and symmetry properties and yielding performance far superior to that of VWTs implemented via other multiwavelets. In the second part of the dissertation, quantization schemes for vector-valued data are studied, and a complete embedded coding system for static vector fields is designed by combining a VWT with suitable vector-valued successive-approximation quantization. Finally, we extend several interframecompression techniques from video-coding applications to vector sequences for the compression of time-varying vector fields. Since the complexity of temporal evolutions of vector features limits the efficiency of the simple motion models which have been successful for natural video sources, we develop a novel approach to motion compensation which involves applying temporal decorrelation to only low-resolution information. This reduced-resolution motion-compensation technique results in significant improvement in terms of rate-distortion performance.