  We have obtained new solutions and methods for the process of thermal
Comptonization. We modify the solution to the kinetic equation of Sunyaev \&
Titarchuk to allow its application up to mildly relativistic electron
temperatures and optical depths $\gtrsim{1}$. The solution can be used for
spectral fitting of X-ray spectra from astrophysical sources. We also have
developed an accurate Monte Carlo method for calculating spectra and timing
properties of thermal-Comptonization sources. The accuracy of our
kinetic-equation solution is verified by comparison with the Monte Carlo
results. We also compare our results with those of other publicly available
methods. Furthermore, based on our Monte Carlo code, we present distributions
of the photon emission times and the evolution of the average photon energy for
both up and down-scattering.
