The Cosmic Microwave Background (CMB) has been detected in 1964 by Penzias and Wilson. It shows today a remarkable constant temperature of T ~ 2.7 K independent of the direction. Present density is about 370 photons per cubic-cm. The size of the hot spots, which deviates only in the fifth decimal of the temperature from the average value, tells us, that the universe is flat. About 300 000 years after the Big Bang at a temperature of T = 3000 K already in the matter dominated era the electrons combine with the protons and 4He and the photons move freely in the neutral universe. So the temperature and distribution of the photons give us information of the universe 300 000 years after the Big Bang. Information about earlier times can, in principle, be derived from the Cosmic Neutrino Background. The neutrinos decouple already 1 second after the Big Bang at a temperature of about 10^{10} K. Today their temperature is ~ 1.95 K and the average density is 56 electron-neutrinos per cubic-cm. Registration of these neutrinos is an extremely challenging experimental problem which can hardly be solved with the present technologies. On the other hand it represents a tempting opportunity to check one of the key element of the Big Bang cosmology and to probe the early stages of the universe evolution. The search for the cosmic relic neutrinos with the induced beta decay Electron-neutrino + 3H --> 3He + e- is the topic of this contribution. The signal would show up by a peak in the electron spectrum with an energy of the neutrino mass above the Q value. We discuss the prospects of this approach and argue that it is able to set limits on the relic neutrino density in our vicinity.