All existing positive results on two-neutrino double-$\beta${} decay in different nuclei were analyzed. Using the procedure recommended by the Particle Data Group, weighted average values for half-lives of Ca48, Ge76, Se82, Zr96, Mo100, Mo100-Ru100 (01+), Cd116, Te130, Nd150, Nd150-Sm150 (01+), and U238 were obtained. Existing geochemical data were analyzed, and recommended values for half-lives of Te128, Te130, and Ba130 are proposed. Given the measured half-life values, nuclear matrix elements were calculated. I recommend the use of these results as the most currently reliable values for the half-lives and nuclear matrix elements.

All existing positive results on two-neutrino double beta decay and two-neutrino double electron capture in different nuclei have been analyzed. Weighted average and recommended half-life values for ${}^{48}$Ca, ${}^{76}$Ge, ${}^{82}$Se, ${}^{96}$Zr, ${}^{100}$Mo, ${}^{100}$Mo - ${}^{100}$Ru (0₁⁺), ${}^{116}$Cd, ${}^{128}$Te, ${}^{130}$Te, ${}^{136}$Xe, ${}^{150}$Nd, ${}^{150}$Nd - ${}^{150}$Sm (0₁⁺), ${}^{238}$U, ${}^{78}$Kr, ${}^{124}$Xe and ${}^{130}$Ba have been obtained. Given the measured half-life values, effective nuclear matrix elements for all these transitions were calculated.

The double-beta-decay experiment NEMO-3 has been taking data since February 2003. The aim of this experiment is to search for neutrinoless (0 νββ) decay and investigate two neutrino doublebeta decay in seven different isotopically enriched samples (¹⁰⁰Mo, ⁸²Se, ⁴⁸Ca, ⁹⁶Zr, ¹¹⁶Cd, ¹³⁰Te, and ¹⁵⁰Nd). After analysis of the data corresponding to 3.75 yr, no evidence for 0 νββ decay in the 100Mo and 82Se samples was found. The half-life limits at the 90% C.L. are 1.1 × 10²⁴ and 3.6 × 10²³ yr, respectively. Additionally for 0 νββ decay the following limits at the 90% C.L. were obtained, >1.3 × 10²² yr for ⁴⁸Ca, >9.2 × 10²¹ yr for ⁹⁶Zr, and >1.8 × 10²² yr for 150Nd. The 2 νββ decay half-life values were precisely measured for all investigated isotopes.

The main achievements during 75 years of research on double beta decay have been reviewed. The existing experimental data have been presented and the capabilities of the next-generation detectors have been demonstrated.

Double beta decay of ${}^{150}\mathrm{Nd}$ and ${}^{148}\mathrm{Nd}$ to the excited states of daughter nuclei have been studied using a $400\hspace{0.5em}{\mathrm{cm}}^3$ low-background HPGe detector and an external source consisting of 3046 g of natural ${\mathrm{Nd}}_2{\mathrm{O}}_3$ powder. The half-life for the two-neutrino double beta decay of ${}^{150}\mathrm{Nd}$ to the excited $0_1^{+}$ state in ${}^{150}\mathrm{Sm}$ is measured to be $T_{1/2}=[1.33{}_{-0.23}^{+0.36}(\mathrm{stat}){}_{-0.13}^{+0.27}(\mathrm{syst})]\times {10}^{20}$ yr. For other $(0\nu +2\nu )$ transitions to the $2_1^{+}$, $2_2^{+}$, $2_3^{+}$, and $0_2^{+}$ levels in ${}^{150}\mathrm{Sm}$, limits are obtained at the level of $~(2\text{−}8)\times {10}^{20}$ yr. In the case of ${}^{148}\mathrm{Nd}$ only limits for the $(0\nu +2\nu )$ transitions to the $2_1^{+}$, $0_1^{+}$, and $2_2^{+}$ excited states in ${}^{148}\mathrm{Sm}$ were obtained and are at the level of $~(4\text{−}8)\times {10}^{20}$ yr.

Two neutrino double beta decay of ¹⁵⁰Nd to the first 0⁺ excited state in ¹⁵⁰Sm is investigated with the 400 cm³ low-background HPGe detector. Data analysis for 11320.5 h shows the excess of events at 333.9 and 406.5 keV. This makes it possible to estimate the half-life of the investigated process as [1.4 _−0.2 ^+.04 (stat)±0.3(syst)]×10²⁰yr.