  An overview of modern experiments on the search for neutrinoless double decay
is presented. The obtained limits on the effective mass of the Majorana
neutrino $\langle m_{\nu} \rangle$ are discussed taking into account the
uncertainties in the value of the nuclear matrix elements (NMEs) and the value
of the axial-vector constant $g_A$. Predictions for the values of $\langle
m_{\nu} \rangle$ from the results of oscillation experiments and modern
cosmological data are presented. The possibilities of the next generation
experiments with sensitivity to $\langle m_{\nu} \rangle$ at the level of
$\sim$ 10-50 meV (studying mainly the inverted ordering (IO) region) are
discussed. %Description of the most developed and promising projects is
presented. The prospects for studying the normal ordering (NO) region are
discussed too. It is shown that the possibilities of studying the NO depend on
the mass of the lightest neutrino m$_0$. In the limiting case of small mass
(m$_0$ $\le$ 0.1 meV), the values of $\langle m_{\nu} \rangle$ $\approx$ 1-4
meV are predicted, which makes the study of this region inaccessible by the
next generation experiments. But there is an allowed region of m$_0$ (7-30 meV)
in the framework of NO, where the predicted values for $\langle m_{\nu}
\rangle$ could be $\sim$ 10-30 meV and that is quite achievable for the next
generation experiments. The possibility to rich in the future sensitivity to
$\langle m_{\nu} \rangle$ at the level of $\sim$ 1-10 meV is also discussed.
