The relationships that connect potential energy surfaces to quantum observables can be complex and nonlinear. In this paper, an approach toward globally representing and exploring potential-observable relationships using a functional mapping procedure is developed. Based on selected solutions of the Schrodinger equation, it is demonstrated that an observable's behavior can be learned as a function of the potential and any other variables needed to specify the quantum system. Once such a map for the observable is in hand, it is available for use in a host of future applications without further need for solving the Schrodinger equation. As formulated here, maps provide explicit information about the global response of the observable to the potential. In this paper, we develop the mapping concept, estimate its scaling behavior (measured as the number of times the Schrodinger equation must be solved during the learning process), and numerically illustrate the technique's globality and nonlinearity using well-understood systems that demonstrate its capabilities. For atom-atom scattering, we construct a single map capable of learning elastic cross sections (i.e., differential cross sections at 2 degrees intervals over angle, as well as integral, diffusion, and viscosity cross sections for scattering energies between 50 meV and 2 eV) involving collisions between any pair of atoms from the Periodic Table. The map for each class of cross sections over the Periodic Table is quantitative with prediction errors shown to be <<1%. We also consider a (3)Sigma (+)(u) Na-2 and create a rovibrational spectral map that encompasses all of the currently proposed potentials for that system. The Na-2 map is highly accurate with the ability to predict rovibrational spectra with errors less than 1x10(-3) cm(-1) over variations in the potential that exceed 130 cm(-1).