After briefly introducing the Q-curvature (roughly speaking a higher dimensional and higher order version of the Gaussian curvature), we study metrics on R^{2m} which are conformal to the Euclidean metric and have constant Q-curvature. This amounts to studying the equation

(-\Delta)^m u e^{2mu} in R^{2m}

As we shall see the dimension of the space plays a crucial role. This problem is important because it leads to compactness and quantization results in the prescribed Q-curvature problem, but also in the mean-field equation or in the Adams-Moser-Trudinger embedding.