A new analytical formalism for studying strong three-body interactions will be presented with a focus on applications. These include evolution of orbital elements during flybys, capture cross-sections, stability boundaries, decay timescales for long-lived unstable triples, orbital elements of decay products, Heggie's law, binary-binary and higher-order interactions. The techniques may be used in large parameter space studies with CPU savings of more than 100,000 compared to direct integrations (for example, flybys past debris disks). They may also be incorporated in traditional N-body codes thereby avoiding bottlenecks associated with small-N subsystems. Particular attention will be paid to the question of stability of arbitrary three-body configurations.