The Fifa president will be limited to three terms in office, in a raft of reforms passed by world football's governing body as it aims to recover from a corruption crisis.
A new council will replace the current executive committee, featuring a female representative from each confederation.
Later on Friday, Fifa will elect a new president to succeed Sepp Blatter.
Four candidates remain in the running after Tokyo Sexwale withdrew at the extraordinary congress in Zurich.
The remaining candidates are Sheikh Salman bin Ebrahim al-Khalifa, Gianni Infantino, Prince Ali bin al-Hussein and Jerome Champagne.
Swiss Blatter, who held the position since 1998, was re-elected for a fifth term last May but resigned amid a growing corruption crisis.
The election process began at 12:00 GMT, but several rounds of voting may be required before a winner is known.
"We stand united in our determination to put things right, so that the focus can return to football once again," said acting Fifa president Issa Hayatou.
"The hard work of restoring trust and improving how we work begins now.
"This will create a system of stronger governance and greater diversity that will give football a strong foundation on which to thrive. And it will deter future wrongdoing."
What are the reforms?
Disclosure of salaries
This will happen on an annual basis for the Fifa president, all Fifa council members, the secretary general and relevant chairpersons of independent standing and judicial committees.
Presidents limited to three terms of four years
This applies to the Fifa president, Fifa council members and members of the audit and compliance committee and of the judicial bodies. Sepp Blatter served five terms as Fifa president dating back to 1998.
Separation of political and managerial functions
The elected Fifa council will replace the executive committee and will be responsible for setting the organisation's overall strategic direction. The general secretariat will oversee the operational and commercial actions needed to implement the strategy.
Promotion of women in football
A minimum of one female representative will be elected as a council member per confederation.
Human rights enshrined in Fifa statutes
Why are reforms needed?
There have been widespread allegations of corruption within Fifa, the arrest of leading officials, the banning of its president and major sponsors withdrawing their support.
Numerous Fifa officials have been indicted in the United States, while Swiss authorities are also investigating the organisation.
Blatter has also been banned from all football activity for six years after being found guilty of breaching Fifa's ethics rules over a $2m (£1.3m) "disloyal payment" to the head of European football's governing body Uefa, Michel Platini.
Former France captain Platini was also suspended. Both men deny any wrongdoing and are appealing to the Court of Arbitration for Sport.
Furthermore, Jerome Valcke, Fifa's secretary general and formerly Blatter's right-hand man, was banned for 12 years following allegations - which he denies - of misconduct while in office.
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