After months of courtship and lower level contacts, the Prime Minister will
meet Colonel Gaddafi, formerly seen by Britain as the greatest state sponsor of terrorism.
But the Conservatives are dead against the meeting: if Blair meets Gadaffi, he should sup with a very long spoon, one of them warned yesterday.
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Further details from BBC News Online
The Tripoli visit follows Libya's decision last December to renounce weapons of mass destruction.
Thursday's trip comes one day after Mr Blair attended the state memorial service for Madrid's bomb victims.
The British prime minister's visit has been criticised by politicians and relatives of some of the 270 people killed in the Lockerbie bombing.
Mr Blair has defended his move, saying he was offering "our hand in partnership" to states giving up terror and banned weapons.
BBC political correspondent Guto Harri says the meeting was one of Mr Blair's "boldest diplomatic manoeuvres ever".
He said that as well as an offer of help training military personnel, which could involve Libyan officers coming to the prestigious British academy in Sandhurst, Libya will also be hoping for key UK backing to ease international restrictions.
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It does not mean forgetting the pain of the past but it does mean
recognising it's time to move on
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British firms are expected to be keen to exploit business opportunities in Libya.
Our correspondent says "he'll shake hands with a man who was seen for decades as a personal incarnation of that terrorist threat."
They are set to shake hands in a large bedouin tent, somewhere on the outskirts of Tripoli.
It is the first visit by a British prime minister since Libya's independence in 1951.
Mr Blair said of his visit: "Let us offer to states that want to renounce terrorism and the development of weapons of mass destruction our hand
in partnership to achieve it, as Libya has rightly and courageously decided to
do.
"That does not mean forgetting the pain of the past but it does mean recognising it's time to move on."
'Timing questionable'
Britain's former ambassador to Libya, Oliver Miles, told BBC Two's Newsnight programme he had doubts about the visit.
"I don't want to say that I think this is a mistake, because I think the key point is that we are right to pursue this process of normalisation and reconciliation.
"The method and the timing are open to discussion."
While it was known that Libya "gave shelter, harbour, encouragement, materials to the IRA, to extremist Palestinian groups, to a whole range of other terrorist and national liberation groups, he hasn't done that in the last 10 years," Mr Miles said.
At prime minister's questions in the House of Commons on Wednesday, deputy Conservative Party leader Michael Ancram told Deputy Prime Minister John Prescott Mr Blair's visit was "highly questionable" and its timing even more so.
Killing unresolved
Britain had suffered from Libyan support for terrorism through Lockerbie, which killed 270 people, the murder of Wpc Yvonne Fletcher and backing for the Provisional IRA, he said.
Foreign Secretary Jack Straw has said that no progress would be made on issues such as investigating Wpc Fletcher's death without engagement with Libya.
The police officer is thought to have been shot dead by a gunman inside the Libyan embassy as she helped police a demonstration outside in 1984.