The incompressible limits of both the n-dimensional (n ≥ 3) Patlak-Keller-Segel (PKS) model and its stationary state are investigated. We derive a geometric free boundary problem of Hele-Shaw type for describing the limit density which relates the compressible (PKS) model and the incompressible (Hele-Shaw) model, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation in the saturated zone. To this end, the novel components are to establish the uniform L 3 estimate of the pressure gradient, the uniform Aronson-Bénilan estimates in L 1 &L 2 &L 3 settings, and the uniform L 1 estimate for the time derivative of pressure. Furthermore, for the Hele-Shaw problem, we prove both the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique, the finite speed of propagation, and the limit energy functional. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.