Wave reflection by smooth media and resonance of systems with radiation damping are instructive examples of a failure of the standard approach to asymptotics. They are also good examples of a type of exponential asymptotics needed for the sciences. Successful modifications of conventional, singular-perturbation theory have been found for them and show some of the principles promising wider usefulness. They have led to recent developments in WKB-connection theory, which are also reported briefly.