The minimization of the free energy of a two-phase system with an interface of arbitrary curvature leads to an extremum (Laplace) condition containing the pressure difference, Delta P, between the two sides of the interface. The expression for Delta P is a function of the normal curvatures and of the resulting bending moments which are themselves functions of the normal curvatures, the mathematical form of which depends on the particular model for the interfacial bending energy that has been employed, On this basis, conclusions can be drawn about the equilibrium shape and curvatures of an interface, e.g., for bicontinuous microemulsions and vesicles. In addition, the pressure difference between the inside and the outside of surfactant-laden interfaces can be calculated. This pressure difference influences the work of formation of microemulsion droplets. A section devoted to the boundary conditions has also been included where in particular the case of a liquid meniscus attached. to a cylindrically shaped solid surface is treated.