We investigate the way the total mass sum of neutrinos can be constrained from the neutrinoless double beta-decay and cosmological probes with cosmic microwave background (CMBR), large-scale structures including 2dFGRS and SDSS datasets. First we discuss, in brief, the current status of neutrino mass bounds from neutrino beta decays and cosmic constraint within the flat ΛCMD model. In addition, we explore the interacting neutrino dark-energy model, where the evolution of neutrino masses is determined by quintessence scalar field, which is responsible for cosmic acceleration. Assuming the flatness of the Universe, the constraint we can derive from the current observation is \(\sum m_{\nu } < 0.87\) eV at 95% confidence level, which is consistent with \(\sum m_{\nu } < 0.68\) eV in the flat ΛCDM model without Lyman alpha forest data. In the presence of Lyman- α forest data, interacting dark-energy models prefer a weaker bound \(\sum m_{\nu } < 0.43\) eV to \(\sum m_{\nu } < 0.17\) eV (Seljark et al). Finally, we discuss the future prospect of the neutrino mass bound with weak-lensing effects.