Elections were held to choose governors in 28 of Nigeria's 36 states and state assembly lawmakers. Governors in the remaining eight states had earlier been chosen in by-elections due to court rulings.
The local elections came three weeks after the candidate of the ruling All Progressives Congress (APC), Bola Tinubu, won a presidential ballot marked by delays and technical problems that rivals said was massively rigged.
According to the results from Saturday's vote declared by the Independent National Electoral Commission (INEC), APC won the governorship in 15 states -- Lagos, Sokoto, Katsina, Jigawa, Gombe, Kwara, Niger, Yobe, Nasarawa, Cross River, Ebonyi, Ogun, Benue, Kaduna and Borno.
The main opposition Peoples Democratic Party (PDP) won seven -- Plateau, Bauchi, Oyo, Delta, Rivers, Akwa Ibom and Zamfara states.
The New Nigeria People's Party (NNPP) emerged victorious in Kano, northern Nigeria's largest state.
More results were expected on Tuesday.
Governors are powerful figures in Nigeria and some control state budgets that are larger than those of several African nations.
The two main opposition presidential candidates -- PDP's Atiku Abubakar, who came second, and Peter Obi of Labour Party who was third -- are challenging Tinubu's victory in court.
The Labour Party said it had filed its official legal petition.
"The process of reclaiming the people's mandate has started," Labour spokesman Yunusa Tanko said in a statement on Tuesday.
In a major upset in the presidential election, outsider Obi won the most votes in Lagos, considered the fiefdom of president-elect Tinubu.
The big question on Saturday was whether Obi's popularity, especially among younger voters, would translate into success at the local polls.
But the APC's Babajide Sanwo-Olu scored a landslide re-election as Lagos governor.
In the key northeastern region of Adamawa -- which could see the election of Nigeria's first woman governor -- the results were declared "inconclusive" as the number of voters unable to cast their ballot was greater than the margin between the two frontrunners.
INEC will set a new date for elections in areas of Adamawa where people had been unable to vote, and afterwards then declare the state-wide result.
President Muhammadu Buhari of the APC steps down in May after two terms, leaving Africa's most populous country grappling with growing insecurity, economic woes and widening poverty.
Nigerians had hoped the presidential ballot would give them a chance to be heard, but many were disappointed by the way the election to replace Buhari was conducted. Turnout for Saturday's local polls was low.
Voters and opposition parties complained last month that technical mishaps with voting machines caused delays and allowed for vote rigging, which the electoral commission has denied.
Local and international observers said the latest poll was impacted by disappointments in the presidential election but also by tactics to scare voters, buy ballots and threats of violence.
"The United States is deeply troubled by the disturbing acts of violent voter intimidation and suppression that took place during the March 18 polls in Lagos, Kano, and other states," the US embassy said in a statement on Tuesday.
EU mission Chief Observer Barry Andrews on Monday said Nigerians' expectations for Saturday's election were not met in many parts of the country.
"Many were disappointed and we witnessed voter apathy that is a clear consequence of failures by political elites and, unfortunately, also by INEC," he said.
His mission also said polling was disrupted by "thuggery and intimidation of voters, polling officials, observers, and journalists."
"Unfortunately, there were many casualties and fatalities," it said. "Vote-buying, also directly observed by EU (election) observers, further detracted from an appropriate conduct of the elections."