Polls opened on time at 0730 GMT in most places but election officials arrived late in some areas. Voting ends at 1330 GMT.
The Independent National Electoral Commission (INEC) is under scrutiny after observers from the European Union, the Commonwealth and other bodies reported several problems during last month's voting, among them failures in systems designed to prevent vote manipulation.
The observers criticised the INEC for poor planning and voting delays but did not allege fraud.
Governors wield enormous influence in Africa's most populous nation of more than 200 million and their support often decides who becomes president. Some governors preside over states whose annual budgets are bigger than some small African countries.
The INEC postponed the gubernatorial poll by a week, saying it needed to reconfigure electronic voting machines that are at the centre of the dispute over the presidential vote won by Bola Tinubu of the ruling All Progressives Congress (APC) party.
The main opposition People's Democratic Party's (PDP) Atiku Abubakar and Labour Party's Peter Obi rejected it as fraudulent and will challenge the results in court.
Voters will choose governors in 28 of 36 states. New state assemblies will be elected in all the states.
EYES ON LAGOS
The race that has generated the most interest is in Lagos, Tinubu's home state and Nigeria's commercial hub, where he is nicknamed the "godfather" for his enduring political influence.
At stake is control of an annual $4 billion budget and running Africa's largest mega city of more than 20 million, home to some of the country's billionaires, including Aliko Dangote who is building a multi-billion dollar oil refining complex.
But Lagos is also teeming with poverty as millions live in slums without power and running water, and residents, rich and poor, have to endure daily traffic jams and pollution.
Tinubu governed Lagos from 1999 to 2007 and has gone on to play a major role in picking every successor since.
Obi, whose support came from young and urban voters, beat Tinubu in Lagos last month. That has buoyed his Labour Party, which is aiming to dethrone APC from running the state.
The ruling APC's incumbent Babajide Sanwo-Olu, who has been in the Lagos state government for the past two decades, faces a strong challenge from Labour Party's Gbadebo Rhodes-Vivour, an architecture and political activist.
The two candidates, as well as Tinubu, voted separately in Lagos.
Gbadebo Rhodes-Vivour alleged voter intimidation on Saturday and said some polling units were moved by the INEC without warning. A spokesperson for INEC did not immediately respond to requests for comment.
Nigerians will also be watching the race in northeastern Adamawa, a conservative and largely Muslim state, which could produce the country's first elected female governor.
There were fewer cases of electoral violence last month but concern still remain about possible clashes in states like Kano in the north and oil-producing Rivers in the south, which have witnessed post-election violence in the past.
(Reporting by MacDonald Dzirutwe, additional reporting by Hamza Ibrahim in Kano; editing by Grant McCool and Ros Russell)
By MacDonald Dzirutwe