They are being taken by boat to the port of Valencia in Spain
The medical charity Médecins Sans Frontières (MSF) say some 629 people have left the rescue boat they were stranded on in the Mediterranean.
They have been transfered to Italian coastguard and navy vessels.
The Aquarius had been instructed by the Maritime Rescue Coordination Centre in Rome to transfer the 629 refugees and migrants so they could be taken to Valencia in Spain to disembark.
It is thought the trip from Italy to Spain will take some three days to complete.
UPDATE: Transfer to Italian coastguard and navy vessels now complete. 51 women, 45 men and 10 children left onboard #Aquarius. While this brings an end to the ordeal of 629 rescued people rescued at sea, it should not set a precedent for future disembarkations in #Mediterranean. pic.twitter.com/dWYmvvQHTq— MSF Sea (@MSF_Sea) June 12, 2018
The rescue ship was left in political limbo after Italy refused to let the vessel dock earlier this week.
The Aquarius had over 600 people on board - including 123 unaccompanied minors, 11 children and seven pregnant women - from six different operations in the central Mediterranean.
The Italian interior minister, Matteo Salvini, reportedly refused to allow it to dock at Italian ports.
MSF say the rescue of two rubber boats "turned critical" when one boat broke apart in the darkness, leaving over 40 people in the water.
After taking 229 people from these boats, the rescue ship then took 400 more people, rescued earlier by Italian navy, Italian coastguard and merchant vessels.
The vessel was told to ask Malta to provide a disembarkation port, but Malta also refused.
The rescue charities insist their role is simply to save lives, and that returning people to Libya is not an option because of the dire security situation in the country.
However, critics claim the presence of rescue charities in the Mediterranean encourages more migrants to attempt the crossing.