In this talk I will review some recent results on blow-up for the classical Patlak-Keller-Segel model of chemotaxis in two-dimensions.
We provide a construction with a proof of nonradial stability for a solution with infinite time blow in the so-called critical mass case. In the supercritical mass scenario we find solutions with multiple blow-up at a single point. The method corresponds to a careful distinction and gluing between inner and outer regimes. These are joint works with Juan Davila, Jean Dolbeault, Monica Musso and Juncheng Wei.