I will briefly introduce the GJMS operators, Q-curvature and their basic properties, then I will describe the possible behaviours of a given sequence of metrics (on a closed Riemannian manifold or on a domain in $R^n$ or similar) having prescribed Q-curvatures which converge to a given function. If the sequence is not-precompact various blow-up phenomena appear, only in part well understood. I will complete the talk with a few open questions.