Yields on 10-year government bonds sank below 1% for the first time as investors bet on an interest rate cut.
Shares in airlines, housebuilders and banks were worst hit, with sharp falls causing a momentary halt in trading.
The falls came after Chancellor George Osborne tried to calm the markets.
In a statement before the financial markets opened, his first since the referendum result, the chancellor said the UK was ready to face the future "from a position of strength".
He also indicated there would be no immediate emergency Budget.
But the upheaval on the financial markets continued.
The pound fell 3.2% to $1.32260, having earlier hit a fresh 31-year low of $1.3151, sinking below the level it had fallen to on Friday when it recorded its biggest one-day fall ever against the dollar. Against the euro, it was down 2.6% at €1.19990.
On the stock markets, the benchmark FTSE 100 index closed down 2.6% at 5982.2.
On Friday the blue-chip index had plunged more than 8% at one point before recovering some ground to close 3.2% lower.
The FTSE 250 index, which mostly contains companies that are more UK-focused, closed down 6.96% on Monday after sliding 7% on Friday. The falls represent the biggest daily percentage falls since 1987.
Stocks on Wall Street also fell. By midday the Dow Jones was down 1.5%, the S&P 500 dropped 1.7% and the Nasdaq lost 2.2%.
Shares in financial firms were the most affected, with Barclays closing down 17.4% and Royal Bank of Scotland plummeting 15.1%.
The dramatic volatility caused trading in both firms to be briefly suspended in early trade.
Property shares were also badly hit, prompting a trading halt, on worries that the decision to leave the EU would hit the housing market.
Easyjet's shares fell more than 22% after the airline said Brexit would contribute to a fall in revenues.
Taylor Wimpey fell 14.9% and Barratt Developments fell over 19%.
Markets in Europe also tumbled, with France's Cac-40 and Germany's Dax both closing down around 3%.
The return on UK government bonds - known as the gilt yield - fell to its lowest level ever on Monday.
Investors are buying more UK government debt in search of safer investments.
The yield from a 10-year gilt dropped below 1% for the first time - hitting 0.993% in morning trading.
The price of gold rose on Monday, up 0.8% to $1,324.60 an ounce, although it was below the two-year peak of $1,358.20 reached on Friday.
The gold price often rises in times of uncertainty as it is viewed as a safe haven asset.
In his statement before the markets opened, Chancellor George Osborne said there would still need to be an "adjustment" in the UK economy, but added it was "perfectly sensible to wait for a new prime minister" before taking any such action.
Mr Osborne's also spoke about the process of the UK's departure from the EU, saying only the UK can trigger Article 50.
"In my judgement, we should only do that when there is a clear view about what new arrangements we are seeking with our European neighbours," he said.
Former Bank of England governor Lord King, who steered the Bank through the 2008 financial crisis, said there was no reason for people to be particularly worried.
"Markets move up, markets move down. We don't yet know where they will find their level and the whole aspect of volatility is that there is a trial and error process going on before markets discover what the right level of stock markets and exchange rates actually are," he told the BBC.
"What we need is a bit of calm now, there's no reason for any of us to panic."
In the short term, Lord King said he expected the leave decision to affect investment from foreign firms into the UK, but in the long term he said it was likely that the impact would be much smaller than either side expected.
However, he criticised the campaigning ahead of the referendum.
"I was struck by how many people said to me that they didn't like the scaremongering tactics. They didn't like to be told that if they were to vote to leave they would be idiots.
"If you say to someone 'you're an idiot if you don't agree with me' then you're not likely to bring them in your direction," he said.
Similarly, Capital Economics economist Julian Jessop said it was important to keep the sharp market movements in perspective, saying it "would be wrong to conclude that the world is on the cusp of another global financial crisis".
"The focus on the magnitudes of the one-day declines obscures the fact that equities had rallied strongly ahead of the UK referendum result in anticipation of a vote to remain in the EU.
"Friday's collapse in the FTSE 100 simply reversed that move, leaving equities little changed over the week."
It was a sentiment echoed by US Treasury Secretary Jack Lew who said the market impact from Brexit had been orderly so far and there were no signs of a financial crisis arising from the vote.
"As we move forward, it is important to stress that UK, European and global policymakers have the tools necessary to support not just financial stability — but also to promote economic growth," he said.
Stock markets plunged on Friday, with more than $2 trillion (£1.5 trillion) wiped off the value of global stock markets, according to Standard and Poor's Dow Jones Indices.
That was the biggest one-day loss in market value - even greater than the value wiped out following the collapse of Lehman Brothers during the 2008 financial crisis, Standard and Poor's calculated.
UK financial markets remain volatile in the wake of the Brexit vote, with sterling plunging to a 31-year low against the dollar, and some share trading temporarily halted.