In Sierra Leone -- the worst affected country -- almost two thirds of people said they had given money to public officials for permits, access to health care and school, according to the "Let the People Have Their Say" report by Afrobarometer. Morocco, Guinea and Kenya were close behind.
To compile the report, Afrobarometer surveyed 51,000 Africans across 34 countries.
The institution rated most corrupt across the whole continent was the police. Alex Vines, head of the Africa Program at Chatham House, said the figures displayed a policing "crisis" in Africa. Nigeria, Kenya and Sierra Leone rated the worst for police corruption.
“If you were to take a group of young people in Africa and say, ‘Someone has burgled your house,’ the majority would not phone the police," he said. "They would rather go to someone else they might know who could sort it out.
“Policing across much of Africa is in crisis. So you get informal police forces in place of the official ones which aren’t doing their job. Vigilante policing provides the protection that the police fail to do.”
In terms of dealing with corruption, the Nigerian and Egyptian governments came out worst in the survey, with 82 percent of people saying it was doing a "fairly bad, or very bad" job tackling the problem.
Bribery was said to be least problematic in Namibia, Mauritius, Cape Verde and Botswana, with between just 4 and 6 percent of people in those countries reporting paying a bribe in the past year.
Although market economies are booming in countries such as Kenya, prompted by oil and gas finds, according to the report endemic corruption has a crippling effect on wealth equality and the poor.
It also said that corruption appeared to be bad for democracy, with people who said their country was corrupt also reporting their governments were undemocratic.
"The research suggests African governments need to step up their efforts to curb corruption, in the interests of both reducing poverty and advancing democracy," the report said.