The first space-borne instruments able to detect and characterize extrasolar terrestrial planets, Darwin (ESA) and TPF-C (Terrestrial Planet Finder-Coronograph, NASA), should be launched at the end of the next decade. Beyond the challenge of planet detection itself, the ability to measure mid-infrared (Darwin) and visible (TPF-C) spectra at low resolution will allow us to characterize the exoplanets discovered. The spectral analysis of these planets will extend the field of planetary science beyond the Solar System to the nearby Universe: It will give access to certain planetary properties (albedo, brightness temperature, radius) and reveal the presence of atmospheric compounds, which, together with the radiative budget of the planet, will provide the keys to understanding how the climate system works on these worlds. If terrestrial planets are sufficiently abundant, these missions will collect data for numerous planetary systems of different ages and orbiting different types of stars. Theories for the formation, evolution and habitability of the terrestrial planets will at last face the test observation. The most fascinating perspective offered by these space observatories is the ability to detect spectral signatures indicating biological activity. In this chapter, we review and discuss the concept of extrasolar biosignatures or biomarkers. We focus mainly on the identification of oxygen-rich atmospheres through the detection of O2 and O3 features, addressing also the case of other possible biomarkers and indicators of habitability.