Here, I look at the concept of a biosignature. The presence of certain gases in a planet’s atmosphere may represent signatures of life-forms. Oxygen is of particular interest. However, while finding a high concentration of oxygen would be suggestive of life, it would not be conclusive if abiotic means of maintenance could be envisaged. Since studies of exoplanet atmospheres are in their infancy, I start in the better-known realm of our solar system, and look at the atmospheres of planets without life, to see the extent to which these vary. Both Mars and Venus have more than 95% carbon dioxide. In contrast, the only solar-system moon to have an atmosphere – Saturn’s Titan – has more than 95% nitrogen. Mercury has virtually no atmosphere. No body in our system has a significant fraction of oxygen except Earth, with about 20%. I examine the techniques of spectroscopy and show how they allow us to see the signatures of particular gases. Then I mention some recent exoplanetary results – such as detection of atmospheric sodium. Finally, I look at proposed direct-imaging space telescopes, notably NASA’s HabEx and LUVOIR, which, if approved, will be missions of extraordinary importance.