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This review is dedicated to recent results on the 2d parabolic-ellipticPatlak-Keller-Segel model, and on its variant in higher dimensions where thediffusion is of critical porous medium type. Both of these models have acritical mass $M_c$ such that the solutions exist globally in time if the massis less than $M_c$ and above which there are solutions which blowup in finitetime. The main tools, in particular the free energy, and the idea of themethods are set out. A number of open questions are also stated.

On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher