For decades it was not clear if neutrinos are massive or just have no mass. It took physicists into this century until they unambigously announced, that neutrinos are indeed massive. This is based on the nobel-prize awarded finding that neutrinos oscillate, i.e. the three different neutrino flavor, can turn into each other. An unavoidable theoretical prerequisite for this phenomen called neutrino oscillations is that the neutrino flavors have a mass difference. Thus, they must be massive, but on the other hand neutrino oscillation experiments doesn't tell the absolute mass scale. But it is the absolut mass scale, which drives the impact of massive neutrinos for elementary particle physics and cosmology!
A closer look on the mass pattern of the twelve elementary particles of the Standard model shows the special role of neutrinos (see figure). The twelve different elementary particles are grouped into three generations (or sometimes called families) each with four particles. In each generation lives a neutrino. Excluding in a first step the neutrinos, the members of the second family are significantly heavier than the first family, and the third generation is again significantly heavier than the second. So intuitively, one would expect the masses of the neutrinos to cluster in the same range as their other family members. We do not yet know the absolut masses of the neutrinos, but we have from experiments upper limits. And these limits show clearly that the neutrino masses are at least by a factor of 106 too light to fit into the mentioned picture.
The Standard Model of particle physics, which very precisely describes the present experimental data, offers no explanation for the observed pattern of the particle masses. In particular, it offers no explanation for neutrino masses and neutrino mixing. This finding gives new momentum to an unsolved question in elementary particle physics: How does nature gives mass to particles?
There are many theories beyond the Standard Model, which explore the origins of neutrino masses and mixings. In these theories, which often work within the framework of Supersymmetry, neutrinos naturally acquire mass. A large group of models makes use of the so-called see-saw effect to generate neutrino masses. Other classes of theories are based on completely different possible origins of neutrino masses. Intrestingly, some of these models predict that the masses of the three different neutrino types should all be nearly the same. Other models predict, that the observed mass variation between the non-neutrinos members of the families is also present for the neutrinos i.e. the neutrino flavours differ significant in mass.
An experiment like KATRIN, which answers the question if at least one neutrino flavor is heavier than 0.2 eV/c2, disentangles the hierarchical and degenerated models. This classification would be a milestone towards the question, how nature provides masses to particles.
Most of the universe´s matter density is in the form of -unknown- dark matter or dark energy. One candidate for dark matter -the only known so far- are massive neutrinos. Even with masses as small as 3 eV/c2 they could make up about 20% of the universe´s mass. With a sensitivity down to about 0.35 eV/c2, KATRIN will either detect a neutrino mass of cosmological relevance or exclude (in case of a negative result) any significant contribution of neutrinos to the universe´s matter content and therefore reduce the role of neutrinos in the forming of large cosmological structures.