We present a new scaling approach to describe an arbitrarily dense polymer interface (layer made of overlapping polymer chains). Our analysis is based on a description of the layer in terms of loops and tails. Within a simple scaling model for the behavior of the loops and tails, we are able to relate the features of the interface to the "loop density profile" S(n), defined as the number (per unit area) of loops and tail having more than n monomers. Our theory predicts (as functions of S) the variations of the monomer density, the extension of the layer, the adsorbance (per unit area), and the effective free energy for various conditions ("good" solvent, Theta solvent, or melt). We are able to recover, in an unusual and rather staightforward manner, previously known results on reversibly and irreversibly adsorbed layers; moreover, new insights into the physics of these situations emerge. New results are obtained for interfaces made of grafted chains which adsorb reversibly on the surface. Both quantitative and qualitative corrections to previous theories are obtained.