One of the drivers of Solar System missions and exoplanet observations is the search for astronomical bodies showing signatures of life. This type of search poses several challenges, mostly related to the definition and characterization of life in the context of exobiology. After giving a general introduction on this subject, I will briefly review the physical limits of terrestrial life and discuss the implications of such limits for the potential existence of life in astronomical environments with extreme physical conditions. Next, I will introduce the subject of planetary habitability, highlighting the importance of climate models in this type of research and showing how it is possible to define an index of surface habitability. After presenting a climate model developed for the study of the surface temperature of Earth-like exoplanets (Vladilo et al. 2015, ApJ 804, 50) I will show examples of applications of this model to terrestrial-type exoplanets discovered with the Kepler mission. Finally, I will briefly discuss the conceptual and experimental difficulties related to the detection of biosignatures in the atmospheres of exoplanets.