On Wall Street, the Dow index fell 2.5% as investors worried those problems could hit the global economic recovery.
The UK's FTSE ended down 3.4% to 5,069, its lowest close since July 2010.
But yields on Spanish and Italian bonds fell sharply after the European Central Bank (ECB) intervened to try to stop the eurozone debt crisis spreading.
Bonds are essentially IOUs issued by governments, or companies, to raise cash. Governments issue new bonds to help pay maturing bonds, which is why it is so important that investors continue to buy them - if they do not, governments are unable to pay their outstanding debts.
The ECB's announcement that it intended to buy up government bonds saw the yield on Spanish 10-year bonds fall from more than 6% to about 5.2% - an indication that investors think it is less risky to lend Spain money. Yields on Italian bonds fell by a similar amount.
Tobias Blattner, a former economist at the ECB, said the central bank's intervention had done little to help the crisis of confidence gripping the share markets.
"This reflects the fundamentals that growth is in a very bad situation on both sides of the Atlantic and this is why the ECB's interventions will not change anything," he told the BBC.
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Europe faces a stark choice: to force some of the weaker countries out of the eurozone or for the stronger countries to assume responsibility for the bloc's debts as a whole.”
The FTSE 100's 3.4% fall was a decline of 178 points. It also marked the first time in the index's 27-year history that it has fallen by more than 100 points for four sessions in a row.Share indexes also fell heavily across Europe, with Germany's Dax ending down 5%, while France's Cac lost 4.7%.
Analysts suggest that further austerity measures, which will be needed to tackle high levels of debt in the US and some eurozone countries, could stifle their already weak economic recovery.
These fears were were also reflected in the price of gold and oil.
Gold, which is seen as a safe investment in times of economic uncertainty, jumped to a new record high of $1,697 an ounce.
Meanwhile, the price of oil slipped further, reflecting concerns that weak global growth could lead to a fall in demand. US light crude fell 3% to $84.23 a barrel, while Brent crude lost 2.6% to $106.54.
Announced after the US markets closed on Friday, the ratings agency cited concerns about the size of the country's budget deficit and the acrimonious and protracted battle in Congress to raise the country's debt ceiling at the eleventh hour. It has graded the US at AA+.
The downgrade was heavily criticised by the US administration, with Treasury Secretary Timothy Geithner telling NBC news S&P had shown "terrible judgement" and a "stunning lack of knowledge about basic US fiscal budget maths".
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Many will see the ECB as taking a serious credit risk in bailing out two financially over-stretched governments and as behaving contrary to the rules of prudent central banking”
But China, which is the world's biggest investor in US debt, has told Washington to address its high levels of debt rather than blaming S&P.An editorial in Monday's China People's Daily newspaper, the mouthpiece of the Chinese Communist Party, called on the US not to "become blind to the great risks that a weak greenback could pose to the world's fragile economic recovery by lifting dollar-denominated commodities prices".
"It is time for the US to tighten its belt and solve its structural problems, in order to resume its reputation and restore world confidence," the paper said.
According to some analysts, this is unlikely to happen.
"We had hoped S&P's action would be a wake-up call [to the US administration]," Mohammed Al-Arian, chief executive of investment giant Pimco, told the BBC World Service.
"Instead, what we have seen is more bickering. There is a massive blame game going on in Washington as to who lost America's AAA rating and also a stunning public clash between the US Treasury and S&P.
"All this means is the politics is getting worse rather than better."
More wobbles? The ECB indicated that it would start buying the bonds of eurozone governments in a statement on Sunday, hoping to instil confidence that some of its biggest economies would not default on their debt obligations.
"Thanks to the ECB's intervention, [yields have] collapsed dramatically. I can't remember the last time I saw such a big move down," said Louise Cooper at BGC Partners.
The G7 group of developed countries also issued a statement saying its members were "determined to react in a co-ordinated manner" to preserve financial stability.
Yet Richard Hunter at broker Hargreaves Lansdown said investors would like to see more being done.
"Until the market can get comfort on these matters, there is going to be more volatility."
On Friday, Italian Prime Minister Silvio Berlusconi announced plans to balance the country's budget by 2013, a year earlier than planned, while Spain has also promised to speed up cost-saving measures.
Yet David Jones, an analyst at financial spread betting firm IG Index, told the BBC that investors would remain unconvinced, despite the various attempts by leaders and international authorities to reassure the markets.
"It hasn't changed the feeling that politicians both in Europe and in the US are always a few steps behind where the crisis is," he said.
"It is a lack of confidence. Markets still think there is a lot of talk from politicians but not much action.
"They are reacting to the crisis rather than putting anything proactive in place. The political issue is a major reason why the markets have been so weak over the past week".