Israel has vowed to stop a flotilla of six ships carrying aid from reaching the Gaza Strip, sending warships to intercept the convoy, according to reports.
The flotilla set sail from a port in Cyprus on Sunday aiming to break Israel's siege of Gaza by landing there on Monday morning.
However, Israel has said that it will not allow the convoy to reach the coastal Palestinian enclave.
Two other ships were damaged over the weekend, and remain in port in Cyprus.
Al Jazeera's Mohamed Vall, reporting from the flotilla's lead vessel, Mavi Marmara, said the ships finally left port after several false starts.
"People [on board] are relaxing, they are having their supper, very happy the final journey has started," he said.
"We have so many kinds of people here ... all of different creeds, philosophies and politics.
"But they are all united around one goal - they want to break the siege on Gaza and do something about what they call an injustice against the Palestinian people."
The organisers of the fleet, dubbed the Freedom Flotilla, said they might launch a second smaller convoy of boats on Tuesday, which would include the two damaged ships, plus a third that has yet to arrive.
"Now we are thinking of sending a second wave of boats including these two and the Rachel Corrie, which is still en route" from Ireland, Audrey Bomse, an adviser to the Free Gaza Movement, said.
Activists on board
Hundreds of pro-Palestinian activists are on board the flotilla, which aims to reach Gaza in defiance of an Israeli embargo on the territory.
The flotilla was originally made up of nine ships - from Turkey, the UK, Ireland, Greece, Kuwait and Algeria -carrying roughly 10,000 tonnes of aid, including cement, water purification systems and wheelchairs.
It was initially expected that the flotilla would set sail on Saturday, but it was delayed over the weekend due to mechanical problems.
The boats were forced to anchor off the coast of Cyprus this weekend.
Hamas, the de facto rulers of the Gaza Strip, have said that the flotilla was about to make history, sending "a strong message that the blockade imposed on the Gaza Strip ... will be broken".
Nicole Johnston, Al Jazeera's correspondent in Gaza, said there was a great deal of excitement in the coastal enclave about the flotilla.
"There have been preparations going on at the port, drills, a Hamas welcoming committee ... but nobody knows if they will actually see this flotilla," she said.
"But one thing that the people of Gaza do appreciate is the international solidarity that they are feeling.
"It reminds people that they haven't been forgotten by the international community."
Israel said the boats were embarking on "an act of provocation" with the Israeli military rather than providing aid, and that it had issued warrants to prohibit their entrance to Gaza.
It asserted that the flotilla would be breaking international law by landing in Gaza, a claim the organisers angrily denied.
Israel has said that it will intercept the boats and detain those on board in the port of Ashdod.