Phase coexistence is frequently observed in molecular monolayers and bilayers. The free energy per unit length of phase boundaries in these quasi-two-dimensional (2D) systems is known as line tension, and is directly analogous to surface tension in three dimensions. The existence of line tension implies the possibility of 2D capillary phenomena, a fundamentally intriguing possibility. Moreover, line tension has important implications with respect to the formation and stability of nm-scale features in thin films, ranging from lithographically-prepared molecular features in devices (e.g. sensor nanoarrays or molecular electronics) to signaling domains in biological membranes (i.e. lipid rafts). It has been proposed that such nm-scale domains may have important ramifications for budding and/or fusion in bilayer membranes. Various methods have been developed to measure line tension, including observations of domain boundary fluctuations, relaxation dynamics, nucleation rates, and others. The competition between line tension and long-range forces (e.g. electrostatic repulsion or curvature elasticity) can lead to a preferred equilibrium domain size, domain shape instabilities, or even unusual domain morphologies (e.g. stripe phases) near critical points. Since liquid crystalline mesophases are ubiquitous in 2D, it is not unusual for the line tension to be anisotropic; this can lead to non-circular domains exhibiting kinks and/or chirality. Recent efforts have been aimed at controlling line tension by the addition of line-active compounds that are analogous to surfactants potentially leading to the observation of new 2D “capillary” phenomena.