  We consider the socalled Bathnagar-Gross-Krook (BGK) model, an approximation
of the Boltzmann equation, describing the time evolution of a single momoatomic
rarefied gas and satisfying the same two main properties (conservation
properties and entropy inequality). However, in practical applications one
often has to deal with two additional physical issues. First, a gas often does
not consist of only one species, but it consists of mixture a mixture of
different species. Second, the particles can store energy not only in
translational degrees of freedom but also in internal degrees of freedom like
rotations or vibrations (polyatomic molecules). Therefore, here, we will
present recent BGK models for gas mixtures for mono- and polyatomic particles
and the existing mathematical theory for these models.
