The chemotaxis system{ut=Δu−∇⋅(uv∇v),vt=Δv−uv,(⋆) is considered under homogeneous Neumann boundary conditions in the ball Ω=BR(0)⊂Rn, where R>0 and n≥2.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data (u0,v0) satisfying u0≥0 and v0>0, the present paper establishes the existence of a globally defined pair (u,v) of radially symmetric functions which are continuous in (Ω¯∖{0})×[0,∞) and smooth in (Ω¯∖{0})×(0,∞), and which solve the corresponding initial-boundary value problem for (⋆) with (u(⋅,0),v(⋅,0))=(u0,v0) in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (⋆) involving arbitrarily large initial data.