A lithium-rich star is a star with an anomalously high abundance of lithium in its upper layers as revealed by a strong absorption line of neutral lithium (Li I) at 6708 Å. Most lithium stars are carbon stars, but lithium enrichment has also found in some normal late-type giants and T Tauri stars. These objects present astronomers with a puzzle because stars destroy most of their lithium soon after formation. Lithium is consumed at nuclear fusion temperatures and not normally remade. For example, the Sun has a lithium abundance about 100 times less than that of the interstellar medium (ISM). High amounts of lithium would certainly not be expected in aging stars such as red giants. Yet some of these evolved suns show lithium in quantities far higher than that of the Sun, and, in extreme cases, as high as that of the ISM. One possible explanation is the recent infall of a large planet or brown dwarf. Such planet swallowing would provide a sudden fresh supply of lithium (which might last 100,000 years or so) and show up as an excess in the star's spectrum. Although this idea, made all the more likely by discoveries of massive planets in tiny orbits around their host stars, works well in most cases, it struggles when faced with the most extreme examples of lithium overabundance found in objects such as the subgiant halo star BD+23 3912. One way a red giant can make fresh helium is by the transport of 7Be from the core to the envelope where it can be converted to 7Li by electron capture. The only known process for manufacturing 6Li is by spallation involving cosmic rays. But these mechanisms, too, are hard pressed to account for the greatest cases of lithium excess. Possibly a combination of factors is at work or there are lithium-generating mechanisms inside stars that are as yet poorly understood.