We consider kinetic equations with collision kernels of the form

∂tf + v · ∇xf = Q(f, f),

where f ≡ f(t, x, v) is the density in the phase space, and Q is the (quadratic) Boltzmann

or Landau operator.

We present the available results about the smoothness with respect to both x and v variables. Those results are obtained using

1. Results of smoothness for the spatially homogeneous equations

2. Averaging lemmas (more precisely, their variant without Fourier transform in time)

3. A procedure of convolution in the v variable which enables to transfer the smoothness

in the x variable from the averages in v to the whole function Propagation of singularities and regularity are shown to hold for the Boltzmann equation with an integrable cross section, while smoothing effects are proven in the case of the Landau equation. We also discuss the applications of the smoothness properties to the study of the large time behavior of the solutions.