<p>This article comprehensively reviews the dynamics, stability, and applications of the triangular libration points (<i>L</i>₄ and <i>L</i>₅) over more than a century of research. A key finding is the Routh critical mass ratio (~ 0.0385) that fundamentally divides their phase-space structure and linear stability. The linear stability condition is satisfied by most planetary systems but exceeded in some binaries. Analytical and numerical studies have revealed a rich spectrum of periodic and quasi-periodic motions, as well as bifurcation and dynamical connections linking triangular and collinear points. While perturbative forces reduce practical stability regions, they do not prevent long-term bounded motion. This has shifted the engineering focus towards fuel-efficient transfer trajectories and the exploitation of natural dynamics for prolonged operations. Despite limited historical use, the inherent dynamical stability and unique geometrical advantage points of <i>L</i>₄/<i>L</i>₅ make them exceptionally promising future sites for deep-space observatories, communication relays, and fundamental physics missions, presenting new challenges for mission design and control.</p>