- 11 US banks injected $30bn into First Republic to avoid collapse.
- US regulators stepped in at the weekend to ensure that customers at SVB and Signature Bank had full access to their money.
Stock markets have risen after a group of US banking giants stepped in to rescue a smaller regional lender, which had been seen as at risk of failure.
Investors' worries over a crisis in the banking sector were eased after 11 US banks injected $30bn (£24.8bn) into the First Republic.
Recent bank collapses in the US have raised fears over the health of the banking system.
The UK's FTSE 100 share index was up 1% in early trade.
Among the UK banks, shares in Lloyds and Barclays were up 1.3% higher.
Stock markets in France and Germany were up by about 0.6%, and earlier, Japan's Nikkei index had closed 1.2% higher.
The 11 US banks who announced the support said the action reflected their "confidence in the country's banking system".
US financial officials said the move was "most welcome, and demonstrates the resilience of the banking system".
After the failure of two US banks in the past week - Silicon Valley Bank (SVB) and Signature Bank - investors have been worried that other banks could also collapse.
US regulators stepped in at the weekend to ensure that customers at SVB and Signature Bank had full access to their money.
Shares in San Francisco-based First Republic had sunk nearly 70% over the last week, amid fears it would be the next bank at risk of a rush of customers withdrawing their deposits.
But the rescue plan by the 11 banks, led by JP Morgan and Citigroup, boosted stock markets, and shares in First Republic were up more than 20% at one point.
However, there are signs that not all worries have been eased.
Shares in First Republic dropped 20 per cent in after-hours stock market trading after the bank said it was suspending its dividend - its payment to shareholders - "during this period of uncertainty".
Swetha Ramachandran, investment director at GAM Investments, said authorities were moving "proactively".
"What they're trying to do is really ringfence the specific issues around individual isolated banks to stop them from becoming systemic," she told the BBC's Today programme.
"So this is very different to 2008 which was a widespread issue across the banking sector."
On Thursday, US Treasury Secretary Janet Yellen said "the banking system overall is safe and sound", while the vice president of the European Central Bank (ECB), Luis de Guindos, said Europe's banking sector was "resilient".
Europe has not escaped the jitters in the banking sector, due to difficulties at Swiss bank giant Credit Suisse.
Shares in Credit Suisse sank earlier in the week on concerns over its future, before the Swiss National Bank said on Wednesday that it was supplying the bank with up to £44bn in emergency funds.
Credit Suisse shares opened higher on Friday but then fell back. The bank's shares are now down by about 22 per cent since the start of the week.
Central banks around the world have sharply raised borrowing costs over the past year to try to curb the pace of overall price rises, or inflation.
The moves have hurt the values of the large portfolios of bonds bought by banks when rates were lower, a change that contributed to the collapse of Silicon Valley Bank, and has raised questions about whether other firms are facing similar situations.
Jeffrey Cleveland, the chief economist at US asset manager Payden and Regal, said other banks could be caught up in the problem.
"There could be other vulnerabilities... if central banks are intent on continuing to raise interest rates," he told the BBC's Today programme.
"Historically when that happens we do see fragility, we do see problems in the financial system."
Before the turbulence in the banking sector erupted, both the US Federal Reserve and the Bank of England had been expected to raise interest rates further at meetings next week. However, due to recent events, some have speculated these rate rises might be scaled back or even scrapped.
On Thursday, the ECB announced a further increase in interest rates from 2.5 to three per cent.
"For the ECB their core fight right now is against inflation," said GAM Investments' Ramachandran.
"While they're monitoring wider financial market stability, I think their view was that the Credit Suisse issues are idiosyncratic and contained to that specific bank."