The Q-curvature equation, a fourth-order elliptic partial differential equation with a critical exponent, is a prominent class of conformal equations, largely due to its connection with a natural concept of curvature. In light of the significant advances in the existence theory for the Q-curvature equation, in parallel with the Yamabe problem, this talk discusses the existence theory in both the compact and non-compact cases. We will also provide several interesting constructions based on techniques such as gluing and Lyapunov–Schmidt reduction, which shed light on the solution set of this equation.