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We present high-resolution Magellan/MIKE spectroscopy of 42 red giant starsin seven stellar streams confirmed by the Southern Stellar Stream SpectroscopicSurvey (S5): ATLAS, Aliqa Uma, Chenab, Elqui, Indus, Jhelum, and Phoenix.Abundances of 30 elements have been derived from over 10,000 individual linemeasurements or upper limits using photometric stellar parameters and astandard LTE analysis. This is currently the most extensive set of elementabundances for stars in stellar streams. Three streams (ATLAS, Aliqa Uma, andPhoenix) are disrupted metal-poor globular clusters, although only weakevidence is seen for the light element anticorrelations commonly observed inglobular clusters. Four streams (Chenab, Elqui, Indus, and Jhelum) aredisrupted dwarf galaxies, and their stars display abundance signatures thatsuggest progenitors with stellar masses ranging from $10^6-10^7 M_\odot$.Extensive description is provided for the analysis methods, including thederivation of a new method for including the effect of stellar parametercorrelations on each star's abundance and uncertainty.  This paper includes data gathered with the 6.5 meter Magellan Telescopeslocated at Las Campanas Observatory, Chile.

The Southern Stellar Stream Spectroscopic Survey (S5): Chemical Abundances of Seven Stellar Streams