The purpose of this invited paper is to give readers a comprehensive and critical overview on how to extract equivalent-circuit models for GaN HEMTs, which are the preferred devices for high-power high-frequency applications. This overview is meant to provide a practical modeling know-how for this advanced type of transistor, in order to support its development for improving device technology and circuit design. With the aim to broaden knowledge to empower models, experimental results are presented as illustrative examples of the most crucial challenges faced by the microwave engineers in modeling high-power GaN HEMTs. All the relevant aspects are covered, going from linear (also noise) to nonlinear models. The analysis is mainly focused on the modeling of distinctive peculiarities of GaN HEMTs. Particular attention is paid to study the importance of accurately modeling the kink effect in the output reflection coefficient, because of the relatively high transconductance, the peak in the magnitude of the short circuit current-gain, because of the relatively large intrinsic capacitances, and the low-frequency dispersion, because of trapping and thermal effects. Furthermore, to emphasize the key role of accurate device models for a successful circuit design, a practical example of power amplifier is discussed. Copyright © 2015 John Wiley & Sons, Ltd