Neither the Greek government nor its eurozone partners want to see Greece leaving the eurozone - or a "Grexit" as it has become known. But if they cannot agree a way forward it could come to that.
To state the obvious, no country has left before.
There is no formal process in which a country can exit the eurozone so much of it would be improvisation.
On Monday night, Greece rejected a plan to extend its €240bn (£178bn) bailout, describing it as "absurd".
Greece is likely to run out of money if a deal is not reached before the end of February.
"We should extend the credit programme by a few months to have enough stability so that we can negotiate a new agreement between Greece and Europe," Greek Finance Minister Yanis Varoufakis told Germany's ZDF.
German Finance Minister Wolfgang Schaeuble dismissed the Greek proposal, telling broadcaster ZDF on Tuesday evening: "It's not about extending a credit programme but about whether this bailout programme will be fulfilled, yes or no."
So what would the consequences of a Grexit be?
What would happen to Greece?
The previous Greek Prime Minister, Antonis Samaras, warned that living standards could fall by 80% within a few weeks of exit.
Unable to borrow from anyone (not even other European governments), the Greek government would simply run out of euros.
It would have to pay social benefits and civil servants' wages in IOUs (if it pays them at all) until a new non-euro currency can be introduced.
The government would not be able to repay its debts, which now amount to a total of about €320bn (£237bn), most of it owed to European governments and agencies and the International Monetary Fund.
The government would have to impose a freeze on withdrawals and on people taking money out of the country. This could lead to queues of ordinary Greeks trying to empty their bank accounts before they get converted into a new currency worth substantially less than the previous one.
In the longer run, Greece's economy should benefit from having a much more competitive exchange rate.
But the devaluation would not solve underlying problems in the economy, including poor tax collection and a struggle to control government spending.
There is also a real possibility of a surge in inflation.
Tax receipts would probably fall as the economy contracted, so the government might finance spending by printing money.
The likely currency depreciation would also be inflationary. It would make imported goods - which in Greece includes a lot of its food and medicine - more expensive.
What would happen to the wider eurozone?
There is a danger that a Greek departure from the euro might do wider economic damage, but the risk is generally thought to be much reduced since 2012, the last time such speculation was rife.
Actions by the European Central Bank are a key element behind this change.
First of all, there is the ECB's commitment to do "whatever it takes to preserve the euro".
That promise, made by the ECB's President Mario Draghi in July 2012, was later fleshed out to include a commitment to buy the debts of governments whose borrowing costs were affected by fears of them leaving the euro.
The ECB has not acted on that promise, but its existence was enough to calm eurozone financial markets. And the ECB could use this initiative if the fears were to re-emerge in the wake of a Greek exit.
There is also quantitative easing, the programme of buying government debt across the eurozone, announced by the ECB in January.
The scheme does not target financially vulnerable countries, but the expectation has already reduced government borrowing costs, which have stayed quite low for all eurozone countries (as implied by the bond market) except Greece.
Having said all that, if Greece really does go, financial contagion cannot be ruled out.
Nervous depositors in other struggling eurozone countries, such as Spain or Italy, may also move their money to the safety of a German bank account, sparking a banking crisis in southern Europe.
Confidence in other banks that have lent heavily to southern Europe - such as the French banks - could also be affected. The banking crisis could conceivably spread worldwide, just as it did in 2008.
What does it mean for businesses?
Greek businesses would face a legal and financial disaster. Some contracts governed by Greek law would be converted into a new currency, while other foreign law contracts would remain in euros. Many contracts could end up in legal disputes over whether they should be converted or not.
Greek companies who still owe big debts in euros to foreign lenders, but whose main sources of income are converted to a devalued non-euro currency, would be unable to repay their debts.
Many businesses would be left insolvent - their debts worth more than the value of everything they own - and would be facing bankruptcy. Foreign lenders and business partners of Greek companies would be looking at big losses.
In the wider eurozone, businesses, afraid for the euro's future, may cut investment. Faced with a barrage of bad news in the press, ordinary people may cut back their own spending. All of this could push the eurozone into recession.
The euro could lose value in the currency markets, providing some relief for the eurozone by making its exports more competitive in international trade. But the flipside is that imports from the rest of the world would become more expensive - especially the US, UK and Japan.
What about political consequences?
If Greece leaves, it undermines the idea that the euro project is irreversible and could give a boost to anti-euro and anti-European Union political forces in other countries.
In Spain, the left-wing anti-austerity party Podemos is already gaining ground, ahead of elections later this year. In Portugal, there is growing fatigue with austerity, and it also goes to the polls this year.
Under European law as it stands, abandoning the euro probably also means leaving the European Union. A lawyer at the European Central Bank wrote in 2009 that "withdrawal from EMU without a parallel withdrawal from the EU would be legally impossible".
But there would be a political aspect to the decision so perhaps some way of keeping Greece in the EU would be found, if all countries involved wanted it enough.