We develop a theory of the double layer at electrolyte I electrolyte interfaces with account for the finite thickness of the interfacial region. This includes the distribution of ions between the two phases and smooth variation of dielectric properties across the interface. The theory offers simple laws for the dependence of the double layer capacitance on the nature of ions, ionic concentrations and potential, which are in line with experimental observations. The theory shows which parameters reflect the nature of ions and the structure of the interface, and how these parameters can be extracted from the capacitance data.