We discuss the evolution of the atmosphere of early Earth and of terrestrial exoplanets which may be capable of sustaining liquid water oceans and continents where life may originate. The formation age of a terrestrial planet, its mass and size, as well as the lifetime in the EUV-saturated early phase of its host star play a significant role in its atmosphere evolution. We show that planets even in orbits within the habitable zone of their host stars might not lose nebular- or catastrophically outgassed initial protoatmospheres completely and could end up as water worlds with CO2 and hydrogen- or oxygen-rich upper atmospheres. If an atmosphere of a terrestrial planet evolves to an N2-rich atmosphere too early in its lifetime, the atmosphere may be lost. We show that the initial conditions set up by the formation of a terrestrial planet and by the evolution of the host star's EUV and plasma environment are very important factors owing to which a planet may evolve to a habitable world. Finally we present a method for studying the discussed atmosphere evolution hypotheses by future UV transit observations of terrestrial exoplanets.