The kinetic Kolmogorov–Fokker–Planck (KFP) equations are the linearized versions for Boltzmann model (non-local integro-differential) or Landau model (local differential). These were introduced by Kolmogorov and were exhibited by Hörmander as the main example for his theory of hypoelliptic equations based on commutator information. But the structure of kinetic spaces is still not well-understood. To draw a parallel, the kinetic spaces are for the KFP equations what the Sobolev spaces are for elliptic equations. They form a scale of spaces in which functions (distributions) $f$ of time $t$, position $x$ and velocity $x$ have regularity along the transport field $\partial_t + v \cdot \nabla_x$ and the velocity field $\nabla_v$, according to the scaling of the equation. We prove that these two informations imply precise regularity in time and position. Our estimates are sharp and scale invariant, improving on earlier works of Hörmander and Bouchut for the regularity with respect to position and in the spirit of the Lions embedding theorem for parabolic equations. They cover the optimal expected range of exponents. The methods rely on Fourier transform estimates. This allows us to give structural information about kinetic spaces (density, completeness and isomorphisms, embeddings…), to precisely describe the distributional solutions to the FKP equations with constant diffusion, to obtain well-posedness for weak solutions when the FKP equations come with rough diffusion and to define their fundamental solutions, and more. This is based on joint works with C. Imbert and L. Niebel.