25 March 2013 Last updated at 17:12 ET
Laiki Bank and Bank of Cyprus will remain shut until later in the week
Most Cypriot banks are to reopen on Tuesday, although the two at the centre of the crisis, Bank of Cyprus and Laiki, will remain shut until Thursday.
The country's president, Nicos Anastasiades said there would be limits on some transactions, but did not give full details.
Capital controls to prevent money leaving the country are already in place.
Certain limits on the size of cash withdrawals are expected to continue.
The banks' reopening came after Cyprus agreed a deal with the International Monetary Fund (IMF) and the European Union (EU) that releases 10bn euros in support.
It was conditional on Cyprus itself raising billions of euros, which it will do by way of a tax on deposits of more than 100,000 euros (£85,000).
The banks shut a week ago after the country's first money-raising solution, which would have hit smaller deposit holders as well as larger holding, was rejected.
Mr Anastasiades said the restrictions would be temporary.
'Specific case'On Monday morning, hopes the deal would solve the crisis lifted shares.
But later stock markets were rocked after the head of the Eurogroup of eurozone finance ministers suggested that the deal for Cyprus model could form a template in any future bailout.
Jeroen Dijsselbloem, the Dutch finance minister who as head of the Eurogroup played a key role in the Cyprus negotiations, said the deal represented a new template for resolving future eurozone banking problems.
"If there is a risk in a bank our first question should be 'OK, what are you in the the bank going to do about that?'," he told Reuters and the Financial Times.
He later added a clarification, saying that Cyprus was "a specific case with exceptional challenges".
The Cyprus deal puts the burden for dealing with problem banks on their shareholders and creditors - in this particular case, customers with large bank balances - rather than the government and taxpayers, or bondholders, who lend through financial markets.
Continue reading the main story “Start Quote
The deal hammered out for Cyprus last night isn't 'fair'. Cyprus has not received the same treatment as other bailed-out eurozone economies”
End Quote The BBC's Andrew Walker points out that the more common approach to failing banks in the current crisis has been for the state to inject new capital.
He says Cyprus's banks are unusual in that they have relatively few financial market investors who could be tapped.
In the past, nations such as Ireland have pumped billions of taxpayers' money into propping up their banks, rather than risk upsetting large investors and spooking the financial system.
Small savers protectedMr Dijsselbloem said the pattern for bank rescues should see shareholders take the first hit, then bondholders, who lend money through financial markets, and only then should depositors with large bank balances.
Continue reading the main story “Start Quote
The bailout and rescue of Cyprus by the eurozone and IMF will not feel like much of a rescue to its people, who face economic misery”
End Quote The Cypriot government suggested that account holders with deposits of more than 100,000 euros should expect to lose about 30% of their balances.
Major depositors, many of whom are wealthy Russians, will not be able to access accounts exceeding the 100,000-euro limit until the restructuring of the banks is complete.
Small savers will be protected but Cyprus's second largest bank - Laiki Bank - will be wound up and split into "good" and "bad" banks, with its good assets eventually merged into the Bank of Cyprus, the country's biggest bank.
The UK's FTSE 100 index ended the day down 0.2%, while Germany's Dax gave up 0.5%, and France's Cac lost 1.1%. In New York, the Dow Jones was 0.5% lower.
In Madrid, the market slipped 2.5% while the Milan index was down 2.27%.
The euro was also driven lower, falling to a six-week low against the pound. The euro was down 0.6% to 84.74 pence.
'Deep recession'Continue reading the main storyJeroen Dijsselbloem
- Aged 46, member of Dutch Labour party
- Became Netherlands finance minister four months ago and took over as chair of Eurogroup in January
- Criticised by predecessor Jean-Claude Juncker and MEPs over initial terms of Cyprus bailout
- No previous experience in finance but studied agricultural economics and business
- Favours reform of the financial sector and a Europe where 'every country brings its budget in order'
The new deal for Cyprus, unlike previous agreements, does not require the approval of the Cypriot parliament.
The uncertainty over the future of Cyprus in the eurozone was sparked a week ago when its parliament rejected an earlier bailout deal, which also included a controversial bank levy.
Despite the Cypriot economy's relatively small size, many analysts had been concerned that the crisis would spread to the wider eurozone, had Cyprus been forced to give up the single currency.
There were fears that the country's possible exit from the euro would trigger a loss of confidence across the single currency bloc, and prompt investors to withdraw from other troubled economies, such as Greece.
However, while Cyprus is now likely to remain in the eurozone, the country still faces significant obstacles as it attempts to recover from the crisis.
The EU-IMF deal involves a massive restructuring of the Cypriot banking system, as well as austerity measures and tax increases.
There has also been significant public anger in Cyprus at the intervention of European authorities, and the credibility of the Cypriot government has been questioned.
"We see a risk that Cyprus' sovereign debt burden post-bailout might not be sustainable, as the country is likely to enter a deep recession caused by the shrinkage of the banking sector and severe deleveraging," warned Reinhard Cluse, an economist at UBS.
