But the latest development, PC Banking from TSB, is the first in the UK to offer real-time banking - showing your balance now, rather than at the previous day's close of business. So if your partner makes a withdrawal from a cash dispenser while you are on line, you will see the transaction as it happens. Julia Roxan, TSB's director of strategy, sums up the service like this: "PC Banking brings the branch into the customer's home. They can pay bills, check their balance and transfer money in an instant, all from their living room."
The service is provided through CompuServe, the on-line information, e-mail and electronic shopping service, and uses elaborate security measures, including data encryption, to protect customers' accounts. Since the service was unveiled two weeks ago, the bank has had more than 2,000 inquiries from potential customers - of which half came by e-mail.
One of the reasons TSB has taken the plunge is that it signs up more customers in the 16-20 age group than any other bank, and many already have their own computers.
But there are one or two downsides. In the interests of security and simplicity, the PC Banking service uses its own software, rather than the normal CompuServe screens.
At present, this is available only to customers with "Wintel" computers - IBM-compatibles which run Microsoft Windows. If your home computer is a Macintosh, you will have to wait for the Mac version - which should arrive later in the year - or use one of the products that run Windows on the Mac, which are expensive. And if you are a teacher or student with an Archimedes, you'll still have to go to the bank in person, rather than electronically.
The next leap is expected to be banking on the Internet, already available to US customers, but not so far in Britain. Security is the big concern, given the global nature of the Internet and the danger of hackers obtaining access to commercial and government systems. But software companies involved with Internet shopping services - which enable credit cards to be used for on-line payment - say that the latest encryption technology has largely overcome this problem.
Many banks and building societies are likely to move into Internet banking. Barclays has its Barclaysquare shopping site, which is one of the most successful launched in the UK, while the Alliance & Leicester's Web pages already offer the ability to submit a mortgage application. But for the majority of us who still go to a real bank, rather than the virtual kind, the biggest chore is undoubtedly bill-paying.
While increasing numbers of bills are paid by direct debit, there are still some that are often easier paid over the counter.
Most credit card accounts allow you to set up a direct debit that will transfer the minimum payment in time for the "payment must reach us by" date shown on your statement. But if you want to pay more - or make an extra payment in mid-month after a shopping spree - the choice, until now has been between posting a cheque or queueing up at the counter. Abbey National is planning to change all that, however, with its new generation of interactive ATMs - cash machines that do much more than display balances and dispense cash.
To pay a bill, simply tap in your PIN and feed the giro form into the machine, which will read the identification of the payee, encoded in the strip across the bottom. You then key in the amount to pay, and the ATM will do the rest, printing out a dated and timed receipt showing whom you have paid and the amount.
The machine will also issue cheques with the payee's name printed on them. Since this a bank cheque, it can be used as near-cash in situations such as legal transactions or high-value purchases where a personal cheque would not be acceptable or would require time to clear. Other services include printing up-to-the-minute detailed statements and setting up automated bill payments by credit transfer.
The new ATMs are currently on trial in three branches - Sheffield, Glasgow, and Dalston, East London. If all goes according to plan, they will roll out across the branch network - but Abbey managers are reluctant to talk about the timing.
Jeanette Hartley, the Abbey manager responsible for the trials, is emphatic that the introduction of these new "smart ATMs" does not signal massive job cuts in the branch network. The aim is to cut down queues - saving customers' time and giving more space for meeting customers.
"It's all about how we cope with growth. We have to take the pressure off the counters. The number of customers is growing, and so is the volume of transactions," she says.
"Many of the products and services we offer nowadays involve sitting down talking with the customer. We can't do that if the branch is packed with people queuing up to get to the counter."
The trials are likely to continue for several months more, testing the hardware, the mix of services and the design of the screens. To make the systems easy to use, the interactive ATMs use touch-screen technology, rather than the traditional push-buttons, and Ms Hartley says: "It's not just a matter of designing the system to undertake the transaction - you have to present it the way customers think it works.
"It may be that some services will appear sooner than others. There are functions that we want to take out across the network, but we can't rush the customers or take them for granted."Reuse content