We study the following distributed source coding problem. Multiple encoders observe correlated sources which they quantize and communicate to a central decoder. The objective is to minimize the distortion as measured by the joint distortion criterion that depends on the sources and the reconstruction. This formulation includes as a special case the problem of computing a function of the sources to within some distortion and also many classic distributed source coding problems. We present a computable inner bound to the optimal rate-distortion region for this problem for the cases when the sources are discrete or jointly Gaussian. When the sources are jointly Gaussian and the joint distortion criterion is such that the decoder is interested in reconstructing a linear function of the sources under mean square error distortion criterion, we present a coding scheme where the encoders perform vector quantization followed by “correlated” binning using good structured nested lattice codes. Next, we consider the general distributed source coding problem for arbitrary discrete sources. We present a new achievable rate-distortion region for this problem based on “good” structured nested random codes built over abelian groups. For certain sources and distortion functions, the new rate regions are strictly bigger than those based on the Berger-Tung coding scheme which has been the best known achievable rate region for the problem till now. Some corollaries of the main coding theorem which are interesting results in themselves are also presented.