What are some of important contributions of the Sumerians to later societies?

The Sumerians were the first group to settle in what we
call Mesopotamia, which means "between two rivers", located between the Tigris and
Euphrates Rivers in modern-day Iraq.  As one of the earliest agricultural settlements,
and earliest "civilizations", the Sumerians contributed much to the
world.

1. Cuneiform.
 Cuneiform is the earliest form of writing.  Cuneiform is made up of a series of wedges
and lines carved into wet clay with a stylus.  The cuneiform script was copied by the
Assyrians, Persians, and Babylonians.  Cuneiform recorded commercial transactions as
well as history.  In essence, we have the Sumerians to thank for writing!
 

2. Irrigation.  The
Sumerians used irrigation for agricultural growth.  Situated between two rivers, the
Tigris and Euphrates, they were able to harness enough water during the hot drought
seasons to survive year round.  One of the unfortunate environmental consequences was
seasonal flooding, so the Sumerians also developed water transport systems to bypass the
northern flood waters.

3.
City-states.  The Sumerians didn't exist in one settlement; they formed
dozens of city-states that served as urban centers.  The city-states often went to war
against one another.

4. Monumental
architecture.  The Sumerians used mud-dried bricks to create architecture
that was used for religious and political purposes.  The ziggurat was a Sumerian pyramid
used for religious purposes; unlike the Egyptian pyramids, ziggurats look more like
stacked boxes (similar to the Mayan and Aztec pyramids).
 

5. Number system based on 10, 60, and
360.  We still use this system today- 60 minutes in an hour, a circle has
360 degrees, etc.