Neutrinos are probably one of the most important structural constituents of the Universe. The Big Bang Theory predicts that the significant component of them is formed by the cosmic neutrino background, an analogues of the big bang relic photons comprising the cosmic microwave background radiation, which has been measured with amazing accuracy. Properties of the relic neutrino background are closely related to the ones of the cosmic microwave radiation. Relic neutrinos pervade space, but their temperature is extremely small, being of the order of 0.1 meV. Although belonging to the most abundant particles of the Universe, the relic neutrinos evade direct detection so far. This is because the low-energy neutrinos interact only very weakly with matter. In this contribution, we explore the feasibility to detect the cosmic neutrino background by means of β-decaying (3H and 187Re) and double beta decaying (100Mo) nuclei. In addition, we address the question whether double relic neutrino capture on nuclei can be an obstacle for observation of neutrinoless double β-decay.