They say wild chimps communicate 19 specific messages to one another with a "lexicon" of 66 gestures.
The scientists discovered this by following and filming communities of chimps in Uganda, and examining more than 5,000 incidents of these meaningful exchanges.
The research is published in the journal Current Biology.
Dr Catherine Hobaiter, who led the research, said that this was the only form of intentional communication to be recorded in the animal kingdom.
She said only humans and chimps had a system of communication where they deliberately sent a message to another individual.
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"They're the only thing that looks like human language in that respect."
Although previous research has found that apes and monkeys can understand complex information from another animal's call, the animals do not appear to use their voices intentionally to communicate messages.
This was a crucial difference between calls and gestures, Hobaiter, who is with Scotland's University of St. Andrews, said.
"It's a bit like if you pick up a hot cup of coffee and you scream and blow on your fingers," she said.
"I can understand from that that the coffee was hot, but you didn't necessarily intend to communicate that to me."
Leaf clipping, for example, where a chimp very obviously takes small bites from leaves is used only to elicit sexual attention.
Many others, though, appear to be ambiguous. A grab, for example, is used for: "Stop that," "Climb on me," and "Move away."
"The big message [from this study] is that there is another species out there that is meaningful in its communication, so that's not unique to humans," said Hobaiter.