We study the global existence of the parabolic-parabolic Keller-Segel system in $$\R^d , d \ge 2$$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $$\tau$$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra & Karch (2015) and Corrias, Escobedo & Matos (2014). Our analysis improves earlier results and extends them to any dimension $$d \ge 3$$. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $$\tau$$, when $$\tau>>1$$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller-Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions.