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Source and channel coding using fountain codes(also known as rate-less erasure codes) have been of much interest during the last years. The capacity achieving property of these codes along with the low de-coding complexity have greatly contributed to these codes becoming more frequently applied in both compression and protection against channel disruptions. One of the major problems of compression and source coding is the Slepian-Wolf problem. The problem gained im-portance because of suggesting that, no matter of seperate or jointly encoding of the sources X and Y, the sufficient rate for reconstruct-ing them remains the same. The report is mostly devoted to fountain code approach of Slepian-Wolf problem. Suitable raptor code design helps us to achieve any arbitrary point of Slepian-Wolf region. Two different points of the Slepian-Wolf region are achieved with being only 3.8 % and 5.4 % off the Slepian-Wolf limit respectively. This fountain code approach is expected to outperform the classical code approach such as Low-Density-Parity-Check(LDPC) codes in terms of decoding complexity and proximity to desired Slepian-Wolf point. In a later part of the thesis, the Belief Propagation (BP) is applied to LT-Markov sub-graphs. Gilber-Eliot (GE) channel is the assumed Markov channel in this work. We come up with the overhead of 16.7% while applying the former Binary Symmetric Channel(BSC) design to LT-Markov sub-graphs. 1