The problem of finding (complete) metrics with constant Q-curvature in a prescribed conformal class is a famous fourth-order cousin of the Yamabe problem. In this talk, I will provide some background on Q-curvature and discuss how several non-uniqueness results for the Yamabe problem can be transplanted to this context. However, special emphasis will be given to multiplicity phenomena for constant Q-curvature that have no analogues for the Yamabe problem, confirming expectations raised by the lack of a maximum principle.