  We present a series of three-dimensional discrete Boltzmann (DB) models for
compressible flows in and out of equilibrium. The key formulating technique is
the construction of discrete equilibrium distribution function through
inversely solving the kinetic moment relations that it satisfies. The crucial
physical requirement is that all the used kinetic moment relations must be
consistent with the non-equilibrium statistical mechanics. The necessity of
such a kinetic model is that, with increasing the complexity of flows, the
dynamical characterization of non-equilibrium state and the understanding of
the constitutive relations need higher order kinetic moments and their
evolution. The DB models at the Euler and Navier-Stokes levels proposed by this
scheme are validated by several well-known benchmarks, ranging from
one-dimension to three-dimension. Particularly, when the local Mach number,
temperature ratio, and pressure ratio are as large as $10^2$, $10^4$, and
$10^5$, respectively, the simulation results are still in excellent agreement
with the Riemann solutions. How to model deeper thermodynamic non-equilibrium
flows by DB is indicated. Via the DB method, it convenient to simulate
nonequilibrium flows without knowing exact form of the hydrodynamic equations.
