LONDON — Greg Stevenson was trying to refinance the mortgage on his four-bedroom home in eastern England when things started going awry. An attempt by his bank, TSB, to shift data to a new computer system had gone spectacularly wrong. For several maddening days, he could not connect to his account, transfer funds or reach anybody at the bank for help.

"I felt abandoned," said Mr. Stevenson, a 31-year-old software developer. "I needed to be moving money around, and I needed access to my bank."

The systems failure in April, affecting nearly two million TSB customers, was a breaking point for Mr. Stevenson. He moved his money to Monzo, a British start-up that is among a growing number in Europe offering checking accounts and A.T.M. cards, but lack physical branches — everything is done through an app.

So-called fintech companies have sought to take on the world's biggest banks for years, but only recently have companies like Monzo begun to build a critical mass. Millions of customers across Europe, most in their 20s or 30s, have signed up over the past two years. And thanks to favorable regulations in the region and an influx of venture capital, that shift is accelerating.

Here in Britain, officials have been concerned about the power of large banks in the wake of the 2008 financial crisis, and they see the start-ups as weakening the hold of traditional lenders. The authorities have adopted policies such as a "regulatory sandbox," allowing what are known as challenger banks to test new financial products and get feedback from regulators before proposing them to customers.

In contrast, while some policymakers in the United States are trying to make it easier to open new banks, progress has been slow. States do not want to cede oversight, and without a license, American financial start-ups must set up partnerships with traditional banks to hold deposits.

Support from regulators in Europe has given momentum to companies entering the market there. In addition to making slicker apps, the companies have slashed fees for spending overseas and wiring money. Last year, Monzo became one of the first challenger banks to receive a license allowing it to hold customers' deposits on its own, a milestone no start-up in the United States has achieved.

"Our regulator is pretty forward thinking," said Tom Blomfield, 33, the co-founder and chief executive of Monzo, referring to Britain.

Finance hasn't been immune to the technological change reshaping many other industries. Consumers and businesses are using new payment services and online lenders, and hedge funds are increasingly relying on complex algorithms.

But Mr. Blomfield — who sat for an interview at his London apartment after walking the office dog, a cockapoo named Bingo — argues that day-to-day retail banking has been relatively unaffected.

Monzo, like other start-up banks, still has to figure out how to be profitable. But it has already begun to reimagine banking for an increasingly cashless society. Customers of the future won't need branches when they can text a customer service representative. Detailed spending breakdowns help account holders control their spending. Doing without retail space and tellers keeps costs down, allowing the company to reduce fees and hire programmers.

"The internet lets you run these traditional businesses at a fraction of the cost," Mr. Blomfield said.

The company sends celebratory GIFs on Twitter to new customers, and offers a bright coral-color A.T.M. card that is Instagram-ready. Users can visit the office to see what's being worked on, test new features and perhaps even pet Bingo. Or they can view the company's product road map, published online.

Monzo got started with crowdfunding and remains a minnow compared with the giants of British banking, but customers — notably younger, wealthier ones — are signing up. Roughly 75 percent of Monzo's 900,000 clients are under 40, split evenly between men and women. On average, they make more than 50,000 pounds, or about $65,000, a year, nearly double the median British salary, and Monzo is adding more than 2,000 customers a day.

Digital banks face innumerable issues, though. Chief among them is how they will make money. A slick app and basic checking accounts are unlikely to be enough, unless they can offer services like mortgages and other loans that come with higher interest rates.

Last year, Monzo's losses quadrupled to £33.1 million. The company is developing a marketplace where customers can shop for financial services offered by other firms, with Monzo collecting a fee. In the meantime, it makes some money from overdraft and A.T.M. charges, and is experimenting with giving out short-term loans of up to £1,000.

As they expand, these start-ups will also face more regulatory scrutiny and targeting by fraudsters. Traditional banks must ensure, for example, that a certain proportion of the deposits they hold are kept in reserve, as a safety net. Such policies help guarantee that a bank will not lose its customers' money, but they make it harder to turn a profit. Now that Monzo holds customer deposits, it must also meet these requirements.

Start-ups also have to fight inertia. People rarely change banks, whether because of an existing mortgage or the headache of updating a Netflix account. Less than a quarter of Monzo's customers deposit their paychecks into their accounts; most use it as an add-on, to take advantage of perks like cheaper foreign exchange transactions. Even Mr. Stevenson, who does his day-to-day banking with Monzo, had to go to another local lender to refinance.

"The big question is, will millions of people switch their primary transaction accounts, where their checks get deposited and where they pay their bills, and do their key transactions," said Hans Morris, a former executive at Citigroup and Visa who now manages Nyca, a New York venture capital fund. "If you're making a historical bet, you would say that they aren't going to switch."

At the same time, traditional banks are adapting. JPMorgan is testing a mobile-focused banking service called Finn, which offers savings and budgeting tools similar to those of Monzo. Goldman Sachs has introduced Marcus, an online savings and lending product that has more than 1.5 million customers.

Mr. Blomfield is confident his company's young customer base will eventually turn to Monzo for other, more lucrative, financial services that it intends to eventually offer.

Monzo plans to announce in coming weeks that it is raising about £100 million, at a valuation of more than £1 billion, according to two people familiar with the deal. The deal will make it one of the biggest start-ups in Europe, and Monzo will use the funds to help enter the United States as early as next year.

Revolut, another banking start-up, has more than 2.75 million customers across Europe, and it is adding 7,000 each day. After raising $250 million in April, it, too, wants to offer services in the United States.

Government policies there are more constraining, however. As a result, most fintech firms in the United States have focused on payment services like Square, and mobile money transfers like Venmo, though companies including Chime have entered partnerships with licensed lenders to offer banking accounts more like Monzo.

Still, investors in Europe have been won over. A record $1.2 billion has been pumped into banking start-ups globally so far this year, 70 percent of which has gone to European companies. That is more than double last year's figure, and a tenfold increase on the amount invested in 2014, according to CB Insights, a market research firm.

Martin Mignot, a partner at Index Ventures and a member of Revolut's board of directors, said TSB's systems failure this year encapsulated the issues traditional banks face: costly retail space and slow, legacy computer systems. Revolut and others are more nimble than their older, larger competitors. "It's a different mind-set," he said.

Even some who once felt that app-based banks were a gimmick have been won over.

"I was a bit skeptical, but I'm definitely a convert," said Sinead Loftus, a legal assistant in Ireland who joined Revolut after growing frustrated with her local bank's fees. "I've convinced my mom to get one, I told all my co-workers to get one, I got my boss one."