Greek banks are closed and will stay shut for the week, after the country's debt crisis took a dramatic turn.
But cash machines are due to reopen in the afternoon and there are reports of fresh queues.
Greece decided on Sunday to shut banks and restrict cash withdrawals after the European Central Bank resolved not to extend emergency funding.
It followed the failure on Friday of talks with Greek creditors on continuing with the bailout programme.
A critical deadline looms on Tuesday, when Greece is due to pay back €1.6bn to the International Monetary Fund - the same day the bailout expires. There are fears of a default and a possible exit from euro.
The French cabinet met on Monday in emergency session. President Francois Hollande said afterwards that a deal was still possible if the Greeks wanted it.
"There are a few hours before the negotiation is definitively closed, in particular for the prolongation of the Greek aid programme."
German Chancellor Angela Merkel's spokesman said that she was "ready for further talks" with the Greek Prime Minister Alexis Tsipras "if he actually wants to".
In its decree bringing in the bank restrictions, the Greek government cited the "extremely urgent" need to protect the financial system due to the lack of liquidity.
The main points are:
- Banks closed till 6 July
- Cash withdrawals limited to €60 (£42; $66) a day for this period
- Cash machine withdrawals with foreign bank cards permitted
- Pension payments not part of capital controls
- Banking transactions within Greece allowed
The crisis came to a head on Saturday after Greece and eurozone countries failed to reach agreement on payment of the last tranche of bailout money.
Mr Tsipras then called a surprise referendum for 5 July on the latest terms offered by Greece's creditors.
In reaction to the crisis, the London, Paris, Frankfurt and Milan stock markets fell sharply in early trading on Monday, following similar falls in Asia.
The euro lost 2% of its value against the the US dollar. Government borrowing costs in Italy and Spain, two of the eurozone's weaker economies, have also risen.
The Athens stock exchange is also closed as part of the measures.
The decree says they were taken as a result of the eurozone's decision "to refuse the extension of the loan agreement with Greece".
Days of turmoil
- Friday evening: Greek prime minister calls referendum on terms of new bailout deal, asks for extension of existing bailout
- Saturday afternoon: Eurozone finance ministers refuse to extend existing bailout beyond Tuesday
- Saturday evening: Greek parliament backs referendum for 5 July
- Sunday afternoon: ECB says it is not increasing emergency assistance to Greece
- Sunday evening: Greek government says banks to be closed for the week and cash withdrawals restricted to €60
Eurozone finance ministers blamed Greece for breaking off the talks, and the European Commission took the unusual step on Sunday of publishing proposals by European creditors that it said were on the table at the time.
But Greece described creditors' terms as "not viable", and asked for an extension of its current deal until after the vote was completed.
The current ceiling for the ECB's emergency funding - Emergency Liquidity Assistance (ELA) - is €89bn (£63bn). It is thought that virtually all that money has been disbursed.
Analysis: Robert Peston, BBC economics editor
The temporary closure of banks in Greece, and the introduction of capital controls, is very bad news for Greece. Greek people will have less money to spend and business less to invest; so an already weak economy will probably return to deep recession.
As for the impact on the rest of the eurozone, corporate treasurers and wealthy individuals will wake up on Monday wondering if their money is safe in the banks of other weaker eurozone economies.
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