  Accurate knowledge of the properties of hydrogen at high compression is
crucial for astrophysics (e.g. planetary and stellar interiors, brown dwarfs,
atmosphere of compact stars) and laboratory experiments, including inertial
confinement fusion. There exists experimental data for the equation of state,
conductivity, and Thomson scattering spectra. However, the analysis of the
measurements at extreme pressures and temperatures typically involves
additional model assumptions, which makes it difficult to assess the accuracy
of the experimental data. rigorously. On the other hand, theory and modeling
have produced extensive collections of data. They originate from a very large
variety of models and simulations including path integral Monte Carlo (PIMC)
simulations, density functional theory (DFT), chemical models, machine-learned
models, and combinations thereof. At the same time, each of these methods has
fundamental limitations (fermion sign problem in PIMC, approximate
exchange-correlation functionals of DFT, inconsistent interaction energy
contributions in chemical models, etc.), so for some parameter ranges accurate
predictions are difficult. Recently, a number of breakthroughs in first
principle PIMC and DFT simulations were achieved which are discussed in this
review. Here we use these results to benchmark different simulation methods. We
present an update of the hydrogen phase diagram at high pressures, the expected
phase transitions, and thermodynamic properties including the equation of state
and momentum distribution. Furthermore, we discuss available dynamic results
for warm dense hydrogen, including the conductivity, dynamic structure factor,
plasmon dispersion, imaginary-time structure, and density response functions.
We conclude by outlining strategies to combine different simulations to achieve
accurate theoretical predictions.
