The limits on transactions, which include caps on withdrawals and money leaving the country, are a first in the 14-year history of the euro.
Across Cyprus, large but orderly lines formed ahead of the opening of banks for six hours from noon, and guards from private security firms reinforced police outside some ATMs and banks in the capital, Nicosia.People filed calmly into banks across the country once they had opened, and no crowd issues were reported.
The country's president, Nicos Anastasiades, expressed his "warm gratitude and deep appreciation towards the Cypriot people for the maturity and spirit of responsibility they have shown at a critical time for the stability of the Cypriot economy," a statement from his office said.
In Nicosia, one 70 year-old pensioner who only gave his name as Ioannis arrived at the bank some two hours ahead of the scheduled opening time.
"I had to come this early, I came from my village 20 kilometers away, what do they want me to do, keep coming and going?" he said.
Banks have been shut since March 16 to prevent people draining their accounts as politicians scrambled to come up with a plan to allow Cyprus to qualify for 10 billion euros ($12.9 billion) in bailout loans for its stricken banking sector.
The deal was finally reached in Brussels early Monday, and imposes severe losses on deposits of over 100,000 euros in the country's two largest banks, Laiki and Bank of Cyprus. Laiki will be broken up, with its good assets being absorbed by Bank of Cyprus. The exact amounts of the losses have not yet been officially announced.
Some individuals and businesses, spotting that Cyprus's economy was in trouble and that a tax on deposits was being discussed, had managed to move their money out Cyprus well before the banks closed their doors last week.
According to European Central Bank figures, deposits in Cyprus's banks slipped 2.2 percent last month, to 46.359 billion euros, the lowest figure since May 2010 and down from a peak of 50.5 billion euros in May 2012. The figure excludes deposits from other banks and the central government.
"I anticipated, not this to happen, but I anticipated issues last year, when Greece had a question of whether it will remain in euro and the consequences of that," said Athos Angelides, who runs a business importing and distributing hair salon products. "So luckily we transferred money in the middle of last year over to the UK."
Although the banks have opened, customers are severely limited in access to their accounts. Capital controls, imposed to prevent worried savers and businesses rushing to withdraw all their money, include limiting cash withdrawals to 300 euros ($383) per day per person and limiting payments abroad to 5,000 euros.
No checks can be cashed, although they can be paid in, and people leaving the country can only take up to 1,000 euros, or the equivalent in foreign currency, with them in cash.
The country's general accounting office said pensions and other social security payments, together with salaries for government employees, will be in bank accounts next Tuesday and Wednesday.
"The Central Bank decided on some limitations, so we are sure that slowly, slowly we are going back to functioning of the banks without serious problems," the head of the parliament, Yiannakis Omirou, told AP.
"Some problems I'm sure will be created but our people are ready to overcome the difficult moments we are passing."
The restrictions will be reviewed daily and are initially in place for seven days until next Wednesday. Some analysts are concerned that, if the controls are kept in place for much longer than that, Cyprus's measures will go against the fundamental principle of the single currency: Free and easy movement of money around the euro group's 17 members.
The European Commission said in a statement that EU member states could restrict financial transactions "in certain circumstances and under strict conditions on grounds of public policy or public security" but added that "the free movement of capital should be reinstated as soon as possible".
Many Cypriots were struggling to work out exactly what they could and couldn't do. Television talk shows hosted dial-ins with experts, with viewers' queries ranging from where they would repay loans to how they could pay tuition fees for children studying abroad and handle check payments. Across the country, people wondered whether they would be able to access their salaries, many of which were due this week.
"I believe this will be a very difficult day for both people and bank employees because no matter how much information there was, things were changing all the time," said Costas Kyprianides, a grocery supplier in Nicosia. "Even us traders, like myself, have so many checks which I need to deposit so I can make ends meet."
During the bank closure, ATMs were working but quickly ran out of money. Those of the two troubled banks, Laiki and Bank of Cyprus, had imposed withdrawal limits of 100 euros a day.
"Up until last night things kept on changing," said store owner Antonis Arotokritou. "There's an overall panic and uncertainty from both the bankers and the rest of the people."
Branches of the country's troubled second-largest lender, Laiki, didn't open on time due to a delay in the bank's computer system. Laiki spokesman Costas Archimandrites said there had been an initial problem with the bank's system but that it was quickly remedied. An hour after the official start of business, all branches had opened.
The stock market announced it would remain closed on Thursday "in order to ensure the smooth functioning of the stock market and protect investors." It too has been closed since March 16.
Elena Becatoros in Nicosia and David McHugh in Frankfurt contributed.