The review of modern experiments on search and studying of double beta decay processes is done. Results of the most sensitive current experiments are discussed. The main attention is paid to EXO-200, KamLAND-Zen, GERDA-I and CUORE-0 experiments. Modern values of T1/2(2ν) and best present limits on neutrinoless double beta decay and double beta decay with Majoron emission are presented. Conservative limits on effective mass of a Majorana neutrino ( < 0.46 eV) and a coupling constant of Majoron to neutrino ( < 1.3 × 10−5) are obtained. In the second part of the review prospects of search for the neutrinoless double beta decay in new experiments with sensitivity to at the level of ∼ (0.01-0.1) eV are discussed. The main attention is paid to experiments of CUORE, GERDA, MAJORANA, EXO, KamLAND-Zen-2, SuperNEMO and SNO+. Possibilities of low-temperature scintillating bolometers on the basis of inorganic crystals (ZnSe, ZnMoO4, Li2MoO4, CaMoO4 and CdWO4) are considered too.