There are a wide assortment of descriptions of the belief propagation algorithm for marginalisation because of its vast applicability. Hence the following thesis aims to use consistent notation first to describe the crux of graphical models, in particular the relationship between Markov random fields, Bayesian networks, and factor graphs. Secondly, to illustrate the fundamentals and preliminary analyses of belief propagation, namely its relevance to Bethe free energy and LDPC codes, and a precursory empirical investigation. Finally, to discuss the application of belief propagation to satisfiability, culminating in survey propagation, one of belief propagation's promising progeny.