Though I personally question all the things attributed as originating with the Sumerians, here are a few things mainstream archeology currently attributes to them. Basically mainstream archeology considers the Sumerians the first permanent society and therefore attributes many wonders to them. Even if first permanent society I think many of these concepts predate them.Formal laws (rule of written law)
First culture to have a written language.
First culture to develop agriculture (plow and irrigation).
Specialized society (people with individual trades/duties)
Wheel used for transportation.
Bookkeeping for trade.
Record keeping for history.
Mathematics (algebra and geometry primarily for astronomy and agriculture) based on base 60 (still use today for clocks, navigation, and geometry (360 degree circle).
Metalworking (smelting and forging)
Architecture (maybe size and scale but suspect predated Sumerians)
Music (lyre, reed pipe, tambourine, drums -- I really suspect several of these were in use before the Sumerians)