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Correlation is often present among observations in a distributed system. This the-sis deals with various design issues when correlated data are observed at distributed terminals, including: communicating correlated sources over interference channels, characterizing the common information among dependent random variables, and test-ing the presence of dependence among observations. It is well known that separated source and channel coding is optimal for point-to-point communication. However, this is not the case for multi-terminal communica-tions. In this thesis, we study the problem of communicating correlated sources over interference channels (IC), for both the lossless and the lossy case. For lossless case, a sufficient condition is found using the technique of random source partition and correlation preserving codeword generation. The sufficient condition reduces to the Han-Kobayashi achievable rate region for IC with independent observations. More-over, the proposed coding scheme is optimal for transmitting a special correlated sources over a class of deterministic interference channels. We then study the genera