We outline the results of a study of the chemical composition of 14 stars in the Sagittarius dwarf spheroidal galaxy (Sgr dSph). For the Sgr dSph stars with [Fe/H >= - 1 the abundances are highly unusual, showing a striking enhancement in heavy s-process elements, increasing with [Fe/H], deficiencies of the alpha-elements (O, Si, Ca, and Ti), deficiencies of Al and Na, and deficiencies of the iron-peak elements Mn and Cu. Our abundances suggest that the composition of the metal-rich Sgr dSph stars is dominated by the ejecta of an old metal-poor population, including products of AGB stars and type la supernovae (SN). We suggest two scenarios to explain the observations: First, is chemical enrichment over long time-scales in a galaxy which has experienced significant mass-loss during its evolution. The second possibility is that we are seeing the products of chemical enrichment from a system which experienced a large burst of star-formation, followed by a quiescent period of many Gyr. Both of these scenarios can lead to the situation where newly synthesized material from AGB stars and type la SN overwhelms nucleosynthesis products from a minor population of young, metal-rich, stars. It is likely that both mechanisms operate in the Sgr dSph. Since these conditions should be generally applicable to low-mass systems, we expect to find similar abundance patterns in other dwarf galaxies. This is supported by the chemical composition of stars in the Galactic globular cluster w Cen and in the Fornax dwarf galaxy, suggesting that both shared a history similar to the Sagittarius dwarf spheroidal galaxy.