  This paper is devoted to the analysis of the classical Keller-Segel system
over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of
the system characterized by various criteria, both in the parabolic-elliptic
case and in the fully parabolic case. The main results when dealing with the
parabolic-elliptic case are: local existence without smallness assumption on
the initial density, global existence under an improved smallness condition and
comparison of blow-up criteria. A new concentration phenomenon criteria for the
fully parabolic case is also given. The analysis is completed by a
visualization tool based on the reduction of the parabolic-elliptic system to a
finite-dimensional dynamical system of gradient flow type, sharing features
similar to the infinite-dimensional system.
