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It will also examine allegations that police and other authorities did not properly investigate prominent offenders such as BBC presenter Jimmy Savile and Cyril Smith, the former Member of Parliament (MP) for Rochdale in northwest England, both of whom are now dead.
"In recent years, we have seen appalling cases of organised and persistent child sex abuse which have exposed serious failings by public bodies and important institutions," said Home Secretary Theresa May in a statement announcing the appointment.
"These failings have sent shockwaves through the country and shaken public confidence in the pillars of society in which we should have total trust."
Woolf, 66, a lawyer and energy market expert, has advised several governments and the World Bank on privatisation and energy reforms.
As Lord Mayor of London, she acts as an ambassador for the city's financial district, a job apart from that of London Mayor Boris Johnson.
She will replace retired judge Baroness Elizabeth Butler-Sloss, who stood down from the position in July.
Critics said she was unsuitable because she is the sister of the late Michael Havers, the government's Attorney-General -- chief legal adviser -- in the 1980s when some of the abuse cases are alleged to have taken place.
Woolf said she was honoured to have been appointed.
"Ensuring lessons are learned from the mistakes which have been made in the past and resulted in children being subjected to the most horrific crimes is a vital and solemn undertaking," she said in a statement.
The precise terms of reference for the inquiry and membership of its panel are yet to be finalised.