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 𝛴𝑚_𝜈 < 0.87 eV at 95% confidence level, which is consistent with 𝛴𝑚_𝜈 < 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 𝛴𝑚_𝜈 < 0.43 eV to 𝛴𝑚_𝜈 < 0.17 eV (Seljark et al). Finally, we discuss the future prospect of the neutrino mass bound with weak-lensing effects.