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.
Only humans and chimps, she said, had a system of communication where they deliberately sent a message to another individual.
"That's what's so amazing about chimp gestures," she told BBC News.
"They're the only thing that looks like human language in that respect."
Although previous research has revealed 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.
BBC Nature - Chimpanzee videos, news and facts
This was a crucial difference between calls and gestures, Dr Hobaiter 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."
Some of the chimps' gestures, the researchers say, are unambiguous - used consistently to convey one meaning.
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."
Although many are very subtle, some of the footage captured by the researchers shows very clearly what the chimps mean to convey.
In one clip, a mother presents her foot to her whimpering offspring, signalling: "Climb on me." The youngster immediately jumps on to its mother's back and they travel off together.
"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 Dr Hobaiter.
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"I don't think we're quite as set apart as we would perhaps like to think we are.
"But then chimps are more closely related to us than they are to the rest of the great apes, so it makes sense that we are incredibly similar to them in many ways."
Dr Susanne Shultz, an evolutionary biologist from the University of Manchester, said the study was commendable in seeking to fill the gaps in our knowledge of the evolution of human language. But, she added, the results were "a little disappointing".
"The vagueness of the gesture meanings suggest either that the chimps have little to communicate, or we are still missing a lot of the information contained in their gestures and actions," she said.
"Moreover, the meanings seem to not go beyond what other less sophisticated animals convey with non-verbal communication.
"So, it seems the gulf remains."

Sum: Researchers say they have translated the meaning of gestures that wild chimpanzees use to communicate.


They say they were not included in negotiations to end the action over pay.
On Friday, state officials threatened to charge more than 700 police with rebellion.
The strike has left a security vacuum, causing a wave of violence that has reportedly left 137 people dead.
Wives and relatives of the officers have been blockading stations because Brazil's military police, who carry out street patrols, are barred by law from going on strike. They face two years in prison if convicted.
Brazil's Defence Minister, Raul Jungmann who was visiting the state, said 3,000 troops drafted in to deal with state security would stay for as long as was needed.
He appealed to the women protesters to allow the police to leave police stations.
"I ask these police wives to not lead your husbands, your partners into a trap. There is no alternative and these actions are not going to lead anywhere."
Similar protests were staged in Rio de Janeiro but most police personnel there did not stop patrols.
Brazil's President Michel Temer addressed the crisis for the first time on Friday calling the strike "illegal" and saying: "The right to protest cannot take the Brazilian people hostage."
The strike paralysed many parts of Espirito Santo with local officials closing schools, clinics and public transport, while shops and businesses remained shuttered.
By Saturday, buses which had stopped running because of concerns that they might be wrecked, had gradually started running again.
In Rio de Janeiro, which has been struggling to pay public sector salaries, family members of police officers organised similar strikes at several police stations but the protest did not stop police patrols in most areas.
Similar protests were attempted in the northern state of Para, in the capital, Belem.
Espirito Santo is one of several Brazilian states struggling with a budget crisis that is crippling public services.
Military police salaries there are the lowest of all 27 Brazilian states.

Sum:
Wives and other relatives of police in the Brazilian state of Espirito Santo have defied a deal aimed at ending a week-long strike.