A detailed analysis has been carried out for a sample of 16 red giants showing a strong Li I 670.8nm line. Ten of them were detected in a survey by Castilho et al. (1998A&AS..127..139C), and the other 6 stars are Li-rich giants selected from the literature. Element abundances in the sample Li-rich giants are similar to those in normal red giants, differing only by their high Li abundance and infrared excess. This suggests that Li-rich giants may correspond to a phase of stellar evolution of normal red giants, when Li is produced and transported to the atmosphere.