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This is a systematic study of the behaviour of finite coverings of (affine)schemes with regard to two Grothendieck topologies: the canonical topology andthe fpqc topology. The history of the problem takes roots in the foundations ofGrothendieck topologies, passes through main strides in Commutative Algebra andleads to new Mathematics up to perfectoids and prisms.  We first review the canonical topology of affine schemes and show, keepingwith Olivier's lost work, that it coincides with the effective descenttopology; covering maps are given by universally injective ring maps, which wediscuss in detail.  We then give a "catalogue raisonn\'e" of examples of finite coverings whichseparate the canonical, fpqc and fppf topologies. The key result is that finitecoverings of regular schemes are coverings for the canonical topology, and evenfor the fpqc topology (but not necessarily for the fppf topology). We discuss a"weakly functorial" aspect of this result.  "Splinters" are those affine Noetherian schemes for which every finitecovering is a covering for the canonical topology. We also investigate theirmysterious fpqc analogs, and prove that in prime characteristic, they are allregular. This leads us to the problem of descent of regularity by(non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology,which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma.

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art