Lehman had incurred losses of billions of dollars in the US mortgage market.
The move threatens to deal a further blow to the global financial system, as banks unwind their deals with Lehman.
Merrill Lynch, also stung by the credit crunch, has agreed to be taken over by Bank of America in a dramatic weekend of events for Wall Street.
Stock markets in Europe and Asia dropped sharply and the dollar tumbled against the euro and the yen as Lehman's failure raised fears about the strength of the global financial system.
The FTSE 100 index of leading UK shares was down 160 points, almost 3%, at 5256.50 in early exchanges.
The global financial economy has never in recent years been tested by quite such a combination of accidents and jolts to confidenceRobert Peston, BBC business editorRead Robert's blogIn quotes: Lehman's demise
Wall Street is also expected to open lower in what is likely to be a tense day of trading.
The chance that Lehman Brothers could collapse increased sharply after the strongest potential buyers pulled out at the weekend.
Barclays and Bank of America had been in talks to rescue the bank.
Greg Wood, the BBC's North America business correspondent, said that police had cordoned off the bank's headquarters in New York and staff were leaving with cardboard boxes as onlookers gathered to watch the bank's demise.
The bank, which employs about 25,000 staff worldwide, including 5,000 in the UK, was founded in 1850 by three brothers.
Lehman Brothers said it intended to file for Chapter 11 bankruptcy protection, which allows a company time to reorganise and devise a plan to pay creditors over time.
It said that its broker-dealer division and asset management division Neuberger Berman Holdings would not be included in the filing.
Bank of America said it had agreed to buy investment bank Merrill Lynch for $50bn (£28bn), in a deal that will create the world's largest financial services company.
Three of the top five US investment banks have now fallen victim to the credit crunch. Lehman and Merrill join Bear Stearns, which acquired for a knockdown price in March by JP Morgan.
The BBC's business editor, Robert Peston, said that it had been Wall Street's most extraordinary 24 hours since the late 1920s.
He said that Merrill's sale was almost as shocking as Lehman's demise.
"The global financial economy has never in recent years been tested by quite such a combination of accidents and jolts to confidence," he said.
In addition to Lehman and Merrill Lynch, problems at AIG, the world's largest insurer, are also mounting.
Reeling from losses on its exposure to real estate, AIG has sought $40bn from the Federal Reserve to shore up its finances, the New York Times has reported.
To help prevent panic on financial markets, the Federal Reserve said for the first time it will accept stocks owned by banks as collateral for short-term cash loans, broadening its emergency lending programme.
Also 10 of the world's biggest banks on Sunday agreed to establish a $70bn emergency fund, with any one of the bank's able to able to tap up to a third of it should they face any liquidity problems.
The Bank of England and the European Central Bank said they were monitoring money markets and stood ready to intervene if necessary.
The Financial Services Authority has temporarily suspended London trading in Lehman Brothers' shares.
