A homogenization framework is developed that accounts for the effect of size at the micro- or nanoscale. This is achieved by endowing the interfaces of the micro- or nanoscopic features with their own independent structure, using the theory of surface elasticity. Following a standard small-strain approach for the microscopic deformation in terms of the macroscopic strain tensor, a Hill-type averaging condition is used to link the two scales. A procedure for determining overall effective properties in the case of composites with elastic components and elastic material interfaces is presented. A special example of multilayered composites demonstrates the correlation between a material interface and a very thin interphase layer.