Tau neutrinos interacting inside the Earth produce τ leptons which thereafter can decay inside the atmosphere. The propagation of extremely energetic ν τ 's and τ's through the Earth is studied by means of a detailed Monte Carlo simulation, taking into account all major mechanisms of ν τ interactions and τ energy loss as well as decay modes. The rates of τ's emerging from the Earth are determined as a function of τ's energy for several cosmic neutrino models.