The various experiments with atmospheric, solar, accelerator, and reactor neutrinos provide compelling evidence that neutrino flavor states are non-trivial superpositions of neutrino mass eigenstates and that neutrinos oscillate from one flavor state into another during flight. From these neutrino oscillation experiments the neutrino mixing matrix U containing the mixing angles as well as the differences between the squares of neutrino masses can be determined .
The values of the neutrino masses are very important for astrophysics and cosmology for describing the role of neutrinos in the evolution of the universe. Although neutrinos are very light they may contribute significantly to the mass density of the universe: with 336 neutrinos per cubic centimeter left over from the Big Bang they are about a billion times more abundant than atoms. Also, the values and the pattern of the neutrino masses are very important for nuclear and particle physics, since they are a very sensitive probe for physics beyond the Standard Model of particle physics at large scales. Since neutrinos are neutral there is the possibility that neutrinos are their own antiparticles and, additionally, so-called Majorana mass terms originating from large scales could play a dominant role in describing neutrino masses .
Clearly, neutrino oscillation experiments prove that neutrinos have non-zero masses, but these experiments – being a kind of interference experiment – cannot determine absolute masses. Therefore, we need other ways to determine the absolute value of the neutrino masses. Three methods are sensitive to the values of the neutrino mass eigenstates and their mixing angles in different ways: neutrino mass from cosmology, neutrino mass from neutrinoless double β-decay and neutrino mass from direct neutrino mass determination. These are described below.
1.1 Neutrino mass from cosmology
The relic neutrinos would have a smeared out fluctuation on small scales, depending on their mass. By analyzing the power spectrum of the universe, limits on the sum of the three neutrino mass states, e.g. eV , have been obtained, which are to some extent dependent on model and analysis.
1.2 Neutrino mass from neutrinoless double β-decay ()
Some even–even nuclei can only decay via double β-decay into a nucleus with higher binding energy. This second-order weak process was proposed more than 70 years ago  and has been experimentally confirmed for around a dozen nuclei over a period of more than 20 years (left-hand panel of Fig. 1). If – in the case of a -decay – the electron antineutrino going out at one vertex is absorbed at the other vertex as a neutrino (right-hand panel of Fig. 1), the double β-decay will be neutrinoless. This would violate lepton number conservation by two units. Therefore, neutrinoless double β-decay is forbidden in the Standard Model of particle physics. It could occur only if the neutrino is its own antiparticle (“Majorana neutrino” in contrast to “Dirac neutrino”). Secondly, the left-handedness of neutrinos and the right-handedness of antineutrinos in charge current weak interactions provide a second obstacle for neutrinoless double β-decay. A finite neutrino mass is the most natural explanation for the production, in the chirality-selective weak interaction, of a neutrino with a small component of opposite handedness on which this neutrino exchange is based. Then the decay rate will scale with the absolute square of the so-called effective neutrino mass, which takes into account the neutrino mixing matrix U:
In the case of neutrinoless double β-decay the neutrino mixing matrix U also contains two so-called Majorana phases in addition to the normal CP-violating phase δ. The latter is important for neutrino oscillation whereas the former do not influence neutrino oscillation but mee. A significant additional uncertainty entering the relation of mee and the decay rate comes from the uncertainties of the nuclear matrix elements of the neutrinoless double β-decay .
In the case of decays there are two alternative processes including one or two electron capture (EC) processes: and ECEC. However, the modes involving positrons are phase-space suppressed and only six possible emitters are known. Since in the case of neutrinoless double β-decay the inner neutrino propagator is not observable, the exchange could be based on a completely different particle allowing this lepton number-violating process, e.g. a particle from theories beyond the Standard Model, which leads to a very interesting interplay with new Large Hadron Collider data , because at the TeV scale the contribution to double β-decay can have a similar amplitude to the light neutrino exchange. Among others there are heavy Majorana neutrinos, right-hand W bosons, and double-charged Higgs bosons, which are getting constrained by measurements of ATLAS and CMS [7-10]. But there is a general theorem that there will always be a Majorana neutrino mass term observed in the case neutrinoless double β-decay . Diagrams like the one shown in the right-hand panel of Fig. 1 are also applicable for other out-going leptons in theories beyond the Standard Model .
There are many recent reviews on neutrinoless double β-decay and neutrinoless double β-decay searches (e.g. [6, 13]).
1.3 Neutrino mass from direct neutrino mass determination
Direct neutrino mass determination is based purely on kinematics or energy and momentum conservation without further assumptions. In principle there are two methods: time-of-flight measurements and precision investigations of weak decays. The former requires very long baselines and therefore very strong sources, which only cataclysmic astrophysical events like a core-collapse supernova could provide. From the supernova SN1987a in the Large Magellanic Cloud, upper limits of 5.7 (95% CL)  or of 5.8 (95% CL)  on the neutrino mass have been deduced, which depend somewhat on the underlying supernova model. Unfortunately nearby supernova explosions are too rare and seem to be not well enough understood to compete with the laboratory direct neutrino mass experiments.
Therefore, the investigation of the kinematics of weak decays and more explicitly the investigation of the endpoint region of a β-decay spectrum (or an electron capture) is still the most sensitive model-independent and direct method to determine the neutrino mass. Here the neutrino is not observed but the charged decay products are precisely measured. Using energy and momentum conservation the neutrino mass can be obtained. In the case of the investigation of a β-spectrum usually the “average electron neutrino mass squared” is determined :
This incoherent sum is not sensitive to phases of the neutrino mixing matrix in contrast to neutrinoless double β-decay.
In β-decay, e.g. , the out-going electron is sharing the decay energy with the out-going electron antineutrino. Therefore the shape of the β-spectrum near its endpoint E0, i.e. the maximum energy of the electron in the case of zero neutrino mass, is sensitive to the neutrino mass as shown in Fig. 2. A recent review of this topic has been published .
1.4 Comparison of the different neutrino mass methods
Figure 3 demonstrates that the different methods are complementary to each other and compares them. It shows that the cosmological relevant neutrino mass scale has a nearly full correlation to determined by direct neutrino mass experiments. The observable of neutrinoless double β-decay, the effective neutrino mass mee, does not allow a very precise neutrino mass determination, e.g. to determine , due to the unknown CP and Majorana phases and the uncertainties of the nuclear matrix elements . In the case of normal hierarchy and small neutrino masses the effective neutrino mass mee can even vanish (left-hand panel of Fig. 3), which is not possible in the case of inverted hierarchy (right-hand panel of Fig. 3). On the other hand, the combination of all three methods gives an experimental handle on the Majorana phases. As already mentioned, in addition the exchange of supersymmetry particles may be the dominant process of neutrinoless double β-decay, which would spoil the complete information on the neutrino mass. Nevertheless, the search for the neutrinoless double β-decay is the only way to prove the Majorana character of neutrinos and one of the most promising ways to search for lepton number violation.
The remainder of this article is structured as follows. Section 'Search for neutrinoless double β-decay' reports on the various searches for neutrinoless double β-decay. In Section 'Direct neutrino mass experiments' the neutrino mass determinations from tritium and β-decay as well as from electron capture are presented. The conclusions are given in Section 'Conclusions'.