It's tricky to pinpoint an exact date for the evolution of eukaryotic cells. Scientists generally believe they emerged around 2.7 billion years ago (bya), but the fossil record is limited and the process was likely gradual.

Here's what we know:

* Earliest evidence: The oldest known eukaryotic fossils are around 2.1 bya, but there's evidence of biomolecules indicative of eukaryotic life dating back to 2.7 bya.

* Endosymbiosis: The most widely accepted theory for the origin of eukaryotic cells is the endosymbiotic theory. This suggests that eukaryotic cells evolved from a symbiotic relationship between prokaryotic cells. A larger prokaryote engulfed a smaller one, leading to the formation of mitochondria and chloroplasts within the eukaryotic cell.

* Gradual evolution: The transition from prokaryotic to eukaryotic cells was likely a gradual process, with many intermediate stages that are difficult to pinpoint in the fossil record.

So, while 2.7 bya is a commonly cited timeframe, it's important to remember that the evolution of eukaryotic cells was a complex process that likely took place over a significant period of time.