We consider the parabolic elliptic Keller-Segel model for bacterial chemotaxis. We concentrate on the three-dimensional case, for which this model arises also as a simplified version of the isothermal Euler-Poisson system modeling stellar dynamics. From the work of Brenner et al. in 1999, it is known that the 3D Keller-Segel model admits an explicit radial imploding self-similar solution. We prove the nonlinear radial asymptotic stability of this blowup profile. For this, we develop a stability analysis framework that applies to a large class of semilinear parabolic equations. In particular, we outline a robust technique to treat the underlying spectral problems. This is joint work with Birgit Schörkhuber (Innsbruck).