Rwandans are voting in a referendum on a constitutional amendment to allow President Paul Kagame to seek a third term in office.
Most voters, some 6.4 million, are eligible to vote on Friday, but around 37,000 Rwandans living overseas were able to have their say on Thursday.
The change would allow Mr Kagame to potentially remain in power until 2034.
Rwandans are expected to vote overwhelmingly in favour of changing the constitution.
Mr Kagame is praised at home and abroad for bringing about development and economic growth, reports the BBC’s Catherine Byaruhanga in the Rwandan capital Kigali.
But his critics accuse him of heavy-handed rule and human rights abuses, she adds.
Mr Kagame himself has said he will wait for the outcome of the referendum before making his decision on whether to run in 2017.
Rwanda’s Senate approved draft constitutional amendments last month allowing Mr Kagame to run for another seven-year term.
But the amendments also shorten the length of a term from seven to five years and maintain a two-term limit.
However, those rules would not come into effect until 2024, after Mr Kagame’s third term.
Mr Kagame could then potentially run for another two five-year terms – ruling for some 40 years.
One 22-year-old voter in neighbouring Uganda’s capital Kampala told AFP: “I came to vote… because we want our president to continue leading us”.
But another said voting was a waste of time “because we know the results already”. “Kagame wants to stay in power… he can have what he wants,” he said.
Mr Kagame’s Rwanda Patriotic Front, an ethnic Tutsi rebel force, ended the 1994 genocide perpetrated by Hutu extremists.
Some 800,000 people – Tutsis and moderate Hutus – are estimated to have been killed.
Donor countries, which support the Rwandan government, have been very critical of the move to change the constitution.
The US urged Mr Kagame to step down in 2017, saying he had “an opportunity to set an example for a region in which leaders seem too tempted to view themselves as indispensable to their own countries’ trajectories”.
The president has hit back at “other nations” for interfering in Rwanda’s internal affairs.
But the issue of African presidents seeking a third term in office has caused unrest elsewhere on the continent.
Violence has engulfed neighbouring Burundi since President Pierre Nkurunziza announced in April his plans to seek a third term, in violation of a peace accord that brought an end to the country’s brutal 12-year ethnic civil war.
And in September, there were major protests in the Republic of Congo as President Denis Sassou Nguesso called a referendum to approve constitutional changes allowing him to stand for a third term.