The objective of this work is to present a systematic study on the overall behavior of composites embedding general interfaces between the constituents. The zero-thickness interface model represents a finite-thickness interphase between the constituents. The term general interface refers to an interface model that allows for both displacement and traction jumps, unlike cohesive or elastic interface models. To set the stage, a comprehensive study on homogenization is carried out to examine the effects of various representative volume elements (RVE) and boundary conditions on the overall response of composites. Next, we extend the homogenization framework to account for interfaces hence, capturing size effects in both particulate and fiber composites. Two new analytical approaches are developed to determine the overall size-dependent response of composites. The first approach extends the composite sphere assemblage (CSA), composite cylinder assemblage (CCA) and the generalized self-consistent method (GSCM) resulting in bounds and estimates on the macroscopic properties of composites. In the second approach, we generalize the Mori{Tanaka method that not only determines the effective properties but also provides the state of the stress and strain in each phase of the medium including the interface. The proposed analytical results are thoroughly verified via a series of numerical examples using the finite element method.