The Standard Model of particle physics describes the properties of the most fundamental constituents of matter – elementary particles. This model has been put to numerous consistency tests and all its major predictions have been experimentally confirmed. Nevertheless it has been firmly established that several observed phenomena do not find their explanation within the Standard Model. Among these phenomena are: the origin of neutrino masses and of neutrino oscillations; a mechanism of violation of matter-antimatter symmetry in the early Universe; and the existence of dark matter. These phenomena mean that the list of fundamental particles will extended one day beyond the 17 particles known today. It is possible that new particles may have masses similar to those of known elementary particles and very weak interaction strength (otherwise they would have been discovered long ago). Dedicated experiments with high intensity of interactions and sensitive detectors are required to discover these feebly interacting particles. Such particles can also be copiously produced in cosmic environments where temperatures and matter densities are enormous as compared to laboratory experiment. This thesis analyses cosmic bounds for a particular class of such feebly interacting particles – sterile neutrinos (or “heavy neutral leptons”). In the first part of the thesis a new mechanism of sterile neutrino production in the interiors of exploding supernovae is considered. This mechanism drastically increases the efficiency of production. Surprisingly this does not lead to stringent bounds on sterile neutrino parameters, given the scarcity of the observational data and many “unknowns” about the details of supernova eexplosion. In the second part of the thesis, the interaction of sterile neutrinos with primeval plasma is analysed. The presence of extra particle species in the primordial plasma changes the dynamics of the Universe and in particular affects the yield of primordial Helium-4 – second most abundant chemical element in the Universe. The thesis discusses a novel effect that arises from sterile neutrino interaction with primordial plasma. This effect significantly changes the existing bounds on the properties of sterile neutrinos that stood untouched for 20 years. Using the same machinery the influence of sterile neutrinos on the global expansion of the Universe is analysed. The limits on the lifetime of sterile neutrinos from the measurements of the anisotropies of Cosmic Microwave Background are derived. It is also demonstrated that while sterile neutrinos can reduce the observed tension between the Hubble constant measurements from late-time and early-time probes, it cannot fully alleviate this tension. The final part of the thesis analyses how the parameter space, accessible to the future intensity frontier experiments changes in view of our new results.