Triple and multiple systems are ubiquitous in nature and arise in many different scales and configurations. In hierarchical triple systems, the inner binary is perturbed by a distant companion. For large mutual inclinations, the Lidov–Kozai (LK) mechanism secularly excites the eccentricity of the inner binary to large values. We analytically study the breakdown of the secular approximation and find that the maximal eccentricity is generally enhanced, allowing closer encounters between the inner binary components, leading to a plethora of possible outcomes. We apply our results to several systems: (1) We find the Hill-stability limit of irregular satellites and reproduce their observed inclination distribution. (2) We show that non-secular evolution could be responsible for the formation of contact binaries such as (2014) MU_69 (Ultima-Thule) from wide binaries in the Kuiper Belt. (3) We show that more Hot-Jupiters should be disrupted from larger distances. (4) We include general relativistic corrections and find a lower bound for the maximal eccentricity in 2.5 Post-Newtonian evolution. We use direct integrations to show that the merger rate of binary Black Holes and Neutron Stars in individual galactic nuclei is greater than previously considered by an order of magnitude, but the final rate is uncertain due to uncertainties in star formation history and supply rate. If time permits I'll discuss the future research directions of multiple interacting systems, including chaotic dynamics, stellar and planetary evolution, and Galactic and cluster tidal potential.