Sumerian, written in cuneiform script around 3200 BCE, is the oldest known written language we can still read. It was used by the ancient Sumerians in Mesopotamia (modern-day Iraq) and represents humanity's first shift from accounting tokens to written symbols.
Cuneiform started as pictographs pressed into clay tablets with a reed stylus — literally wedge-shaped marks (cuneiform means 'wedge-shaped'). Over centuries, the pictures became abstract symbols representing sounds and concepts. The oldest legible texts are inventory lists and temple records, which makes sense: writing was invented to solve a practical problem — tracking grain, livestock, and labor.
Other ancient contenders: Egyptian hieroglyphics emerged around the same time (c. 3200 BCE) but Sumerian texts are slightly older in the archaeological record. Akkadian (another Mesopotamian language, c. 2350 BCE) used the same cuneiform script. Proto-Elamite (c. 3200 BCE) exists but is only partially deciphered. Writing systems in China (oracle bone script, c. 1200 BCE) and the Indus Valley (c. 2600 BCE) are either undeciphered or emerged later.
Pro tip: Sumerian was a 'dead language' for 2,000 years — nobody spoke it — but scribes kept copying it the way Latin was preserved in medieval Europe. This is why we have enough texts to decipher it. The oldest writings in a living language are much younger (Hebrew biblical texts around 800 BCE, Old Tamil around 300 BCE).