The standard Big Bang cosmology predicts that the universe is abundantly populated with neutrinos. As expected there are at least 114 neutrinos per cubic centimeter averaged over the whole space. Like the cosmic background radiation the cosmic neutrinos at present posses a very small kinetic energy due to expansion of the universe. This prediction is one of the cornerstones of modern cosmology. On the other hand the existence of cosmic neutrinos has not yet been confirmed by direct detection experiments. For now we only have a lower limit on the total mass of this free floating ghostly gas of neutrinos, but even so it is roughly equivalent to the total mass of all the visible stars in universe. There could be many more neutrinos at Earth because of condensation of neutrinos, now moving slowly under the gravitational pull of our galaxy. Here we discuss the possibility of detection of relic neutrinos in KATRIN and MARE experiments via neutrino capture on tritium and rhenium, respectively. We also examine single and double relic neutrino capture on double beta-decaying nuclei which might be relevant in the context of the new generation double beta decay experiments. Further we explore feasibility of experiments for detection of heavy sterile neutrinos with masses in MeV region, which may have important astrophysical and cosmological implications.