  We present the current status of the analytic theory of brown dwarf evolution
and the lower mass limit of the hydrogen burning main sequence stars. In the
spirit of a simplified analytic theory we also introduce some modifications to
the existing models. We give an exact expression for the pressure of an ideal
non-relativistic Fermi gas at a finite temperature, therefore allowing for
non-zero values of the degeneracy parameter ($\psi = \frac{kT}{\mu_{F}}$, where
$\mu_{F}$ is the Fermi energy). We review the derivation of surface luminosity
using an entropy matching condition and the first-order phase transition
between the molecular hydrogen in the outer envelope and the partially-ionized
hydrogen in the inner region. We also discuss the results of modern simulations
of the plasma phase transition, which illustrate the uncertainties in
determining its critical temperature. Based on the existing models and with
some simple modification we find the maximum mass for a brown dwarf to be in
the range $0.064M_\odot-0.087M_\odot$. An analytic formula for the luminosity
evolution allows us to estimate the time period of the non-steady state (i.e.,
non-main sequence) nuclear burning for substellar objects. Standard models also
predict that stars that are just above the substellar mass limit can reach an
extremely low luminosity main sequence after at least a few million years of
evolution, and sometimes much longer. We estimate that $\simeq 11 \%$ of stars
take longer than $10^7$ yr to reach the main-sequence, and $\simeq 5 \%$ of
stars take longer than $10^8$ yr.
