Biological experience and intuition suggest that self-replication is an inherently complex phenomenon, and early cellular automata self-replication models developed by computer scientists and mathematicians supported that view. However, since von~Neumann's original work in the 1950's, the study of cellular automata models of self-replicating systems has progressively led to smaller and simpler systems. This thesis demonstrates for the first time that it is possible to create automatically self-replicating structures in cellular automata models rather than, as has been done in the past, to design them manually. These emergent self-replicating structures employ a General Purpose Self-Replicating cellular automata rule set which can support the replication of structures of different sizes and their growth from smaller to larger ones. This thesis also demonstrates that, by letting self-replicating structures carry additional information besides replication instructions, they can be used to solve computationally hard problems such as the Satisfiability (SAT) problem. It is shown that self-replicating structures can be made to carry characteristic codes and selection forces can be implemented in cellular automata space. This study opens the door to further studies that could lead to general, solution-evolvable structures and truly self-programming systems. (Also cross-referenced as UMIACS-TR-96-85)