The lasting achievements and contributions of Mesopotamia are their base 60 system, their writing system (cuneiform), and the creation of the wheel.
These are very significant achievements because we still use them today. We use the base 60 system to tell time because we still have 60 seconds in a minute and 60 minutes in an hour. Their writing system, cuneiform, got adapted to 12 different languages that are still used today. Finally, their creation of the wheel was a huge accomplishment because we use wheels for cars, moving things around, etc.