In 2000, Eliashberg, Givental, and Hofer sketched a new approach called symplectic field theory to construct invariants of contact and symplectic manifolds. Despite an extensive literature by Abreu, Bourgeois, Cieliebak, Colin, Ekholm, Eliashberg, Macarini, Oancea, Pati, Ustilovsky, Vaugon, and M.L. Yau, even cylindrical contact homology, the least complex of these invariants, has yet to be rigorously defined or computed in any non-trivial situation. This paper establishes how the heuristic arguments sketched in the aforementioned literature are not sufficient to define a homology theory. After introducing a class of contact forms which we call dynamically separated, we provide a rigorous foundation for cylindrical contact homology in dimension 3, reliant only upon established analytic techniques of Audin, Cieliebak, Damian, Dragnev, Hofer, Kriener, McDuff, Salamon, Schwarz, Wendl, Wysocki, and Zehnder. We then provide a new aproach to compute cylindrical contact homology for a large class of examples. The issue of invariance under the choice of nondegenerate dynamically separated contact form or choice of compatible almost complex structure remains unresolved.