Coarse-graining (CG) approaches are becoming essential tools in the study of complex systems because they can considerably speed up computer simulations, with the promise of determining properties in a range of length scales and time scales never before possible. While much progress in this field has been achieved in recent years, application of CG methods is still inhibited by the limited understanding of a number of conceptual points that need to be resolved to open up the field of CG to a wide range of applications in material science and biology. In this paper, we present some of the key findings that emerged from the development of the integral equation theory of coarse-graining (IECG), which addresses some of the fundamental questions in coarse-graining. Although the IECG method pertains to the CG of polymer liquids, and specifically homopolymer melts are illustrated here, many of the results that emerge from the study of the IECG approach are general and apply to the CG of any molecular liquid. Through this method, we developed a formal relation between the statistical mechanics of CG and a number of predicted physical properties. On the basis of the theory of liquids, the IECG affords the analytical solution of the intermolecular potential for macromolecules represented by a Markov chain of CG sites, thus providing a transparent tool for analysis of the properties in coarse-graining. We identify three key requirements that render a CG model useful: accuracy, transferability, and computational efficiency. When these three requirements are fulfilled, the CG model becomes widely applicable and useful for studying regions in the phase space that are not covered by atomistic simulations. In the process, the IECG answers formally a number of relevant questions on how structural, thermodynamic, and dynamical properties are modified during coarse-graining. It sheds light upon how the level of CG affects the shape of the CG potential and how, in turn, the shape of the potential affects the physical properties. It tests the validity of selecting the potential-of-mean force as the effective pairwise CG potential and the role of higher-order many-body corrections to the pairwise potential to recover structural and thermodynamic consistency of the CG model. Because the IECG theory can be analytically formalized, it does not suffer from the problem of transferability and, in the canonical ensemble, leads to consistent pair distribution functions, pressure, isothermal compressibility, and excess free energy at variable levels of CG from the atomistic to the ultra-CG model, where macromolecules are represented as interpenetrable soft spheres.