  We present a comprehensive review on the state-of-the-art of the approximate
analytic approaches describing the finite-temperature thermodynamic quantities
of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact
repulsive interactions. This paradigmatic model of quantum many-body-theory
plays an important role in many areas of physics -- thanks to its integrability
and possible experimental realization using, e.g., ensembles of ultracold
bosonic atoms confined to quasi-1D geometries. The thermodynamics of the
uniform Lieb-Liniger gas can be obtained numerically using the exact thermal
Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the
TBA numerical calculations do not allow for the in-depth understanding of the
underlying physical mechanisms that govern the thermodynamic behavior of the
Lieb-Liniger gas at finite temperature. Our work is then motivated by the
insights that emerge naturally from the transparency of closed-form analytic
results, which are derived here in six different regimes of the gas and which
exhibit an excellent agreement with the TBA numerics. Our findings can be
further adopted for characterising the equilibrium properties of inhomogeneous
(e.g., harmonically trapped) 1D Bose gases within the local density
approximation and for the development of improved hydrodynamic theories,
allowing for the calculation of breathing mode frequencies which depend on the
underlying thermodynamic equation of state. Our analytic approaches can be
applied to other systems including impurities in a quantum bath, liquid
helium-4, and ultracold Bose gas mixtures.
