More from Ed Miliband who says: "At least we should have consulted the victims. Let's make sure this doesn't happen again. Let's make sure the victims are properly consulted. Let's get the right person and let's get on with it."

Tom Bateman, BBC Radio 4 Today reporter

tweets:, external "Lucy Duckworth confirms on #bbcpm that NSPCC was only group at abuse inquiry meeting not to call for Woolf to stand down. @PeterWanless."

Matthew Reed, chief executive of The Children's Society, says "this critical inquiry has already been delayed twice. Now, it is vital that, as steps are taken to find a new chair, its work continues. The victims and survivors of child sexual abuse have already waited too long to get the justice they deserve".

Labour leader Ed Miliband says that anybody who "properly consulted" the victims wouldn't have appointed Mrs Woolf. He says Theresa May has some questions to answer given that two heads of the inquiry have now resigned.

tweets:, external "Should Home Office ask Prof Alexis Jay who did Rotherham to head this sex abuse inquiry after Woolf quit? Already an adviser."

Lucy Duckworth, chair of the Minister & Clergy Sexual Abuse Survivors, external group, said the victims' concern was that she lacked experience in child protection.

Ms Duckworth said: "It's never going to be an independent inquiry unless it's a statutory inquiry.

"There is no other option but to make this a statutory inquiry where people are forced to give evidence under oath."

tweets:, external "The @ukhomeoffice should never have put Fiona #woolf in the position they did. Now for goodness sake pause & plan."

tweets, external: "Thing is, who's going to head an Government Inquiry who *isn't* a member of The Establishment? Joey Essex?"

BBC 5 Live's John Pienaar says the inquiry's eventual findings will be met by a public who now have a "great deal of scepticism and doubt" about the inquiry before it even gets started.

tweets:, external "At last! Fiona Woolf, head of the historical child sex abuse inquiry, has fallen on her sword."

BBC 5 Live chief political correspondent John Pienaar says Mrs Woolf's resignation has given Theresa May "not so much a political headache as a splitting migraine".

tweets: , external"The CSA Inquiry needs a chair who can command the respect & confidence of victims. Another false start will further undermine the Inquiry."

tweets, external: "Fiona Woolf has just done the right thing & quit. It is now imperative victims are listened to & someone clearly independent is appointed"

tweets, external: ""Theresa May has some questions to answer. To lose one chair is misfortune, to lose two is total carelessness on her part" - @Ed_Miliband"

tweets, external: "Woolf's resignation another blow to May's undeserved "safe hands" reputation."

Nick Robinson
Political editor

tweets, external: "Is time catching up with the longest-serving Home Secretary for more than 50 years? Theresa suddenly doesn't look so Teflon."

Lawyer Alison Millar, who represents the victim groups, say they feel "disappointed, aggrieved and let down" about how the process has been handled. She said the entire inquiry needed to be "re-booted" with much more consultation with survivors.

tweets, external: "Real questions about Home Secretary Theresa May's competence in now losing a second chair of the CSA inquiry."

In her statement, Theresa May confirms she will appoint a new chair but does not say how quickly. She says Mrs Woolf would have fulfilled the role with integrity, impartiality and to the highest standard and insists that victims of abuse "can have confidence" in the rest of the members of the inquiry panel.

Mr Vaz says the next chairman must be selected "as quickly as possible" after an "open, robust, and vigorous" appointment process, since two heads of the inquiry have stepped aside already.