The benchmark FTSE 100 index jumped 1.59% to 7,382.9 points.
US shares also hit record highs, with the Dow Jones Index pushing through the 21,000 level for the first time.
Some traders attributed the rise to Donald Trump's speech to Congress overnight, while others said it was to do with growing expectations of a US rate rise.
On the currency markets, the pound fell 0.44% against the US dollar at $1.2326 and shed 0.36% against the euro to 1.16664.
Mr Trump pledged $1trn in infrastructure spending and "massive" tax cuts for the middle classes.
However, he did not detail how any of it would be paid for.
Greg McKenna, chief market strategist at AxiTrader, said it was a "bit of a disappointment in the sense that he did not put any more meat on the bones" of past promises.
Connor Campbell, financial analyst at Spreadex, said the rise in the FTSE was mainly to do with currency market moves.
"The weakened pound is what has been helping the FTSE 100 Index higher," he said.
"The US dollar is also strengthening because there is every chance that there is going to be a rate hike from the US Federal Reserve on March 15," he added.
Rental equipment firm Ashtead Group was the biggest riser on the FTSE, up 5.7%, while retailer Next added 4.6% following a broker upgrade.
The broadcaster ITV added 4.5% after announcing a Â£200m special dividend with its annual results.
On the FTSE 250, outsourcing firm Mitie Group added 3.6% after selling its troubled social care businesses for a nominal Â£2 to private equity firm Apposite Capital.
The London stock market closed at a record high, as the pound fell against the dollar and the euro.