We propose some new sufficient conditions for the existence of periodic solutions of an asymmetric oscillator with a positive damping term. Our results are complemented by an example where, in some situations, no periodic solutions may exist. This fact is well known in the undamped case, when the resonance phenomenon may appear. However, the damped case presents some unintuitive features which have not been so thoroughly studied in the literature, and the overall picture still has several aspects which need to be better understood.
