Let's say you are in the UK, and have a student loan. In the UK, student loans are subsidized by the government, and you get really low rates of interest. The current rate is 2.4 percent. (Yes, really. Sorry, USAians.)
Now, say you've gone through university and you've managed to land that dream job. It pays a good amount - far more than you had when you were living as a student. You have this big student loan, and you're finally earning some money and you figure you can pay it all off now - and that would be the smart thing to do, right?
NO! What you should do is pay back the minimum required and invest the rest. Why? Because you can get more interest than they can from you!
Yes, I know - it's a large amount of money and you don't want it weighing on your mind. But you have to think of the opportunity cost. You've been given that money, and 2.4% is a piddly amount of interest. You can do better than that just dumping it in an online savings account (but you should already have been doing that with the funds you got - and you should have taken the maximum loan, since it was such a good deal). Now you're working, you can do even better by putting the amount you've had spent paying more off the debt into a retirement savings account instead - in the UK, that'd be an ISA, while the USA has the IRAs and 401(k)s. Say the investments pay 8%, and you lose 2% to taxes (remember, some investments are tax free). The difference between what you're getting minus what you're paying (and minus taxes) is profit for you. You just made 8 - 2.4 - 2 = 3.6% on that money.
This doesn't just apply to student loans. If you have a mortgage (especially a low fixed-rate mortgage) then in many cases it makes far more sense to invest in retirement through a Roth IRA or your company's 401(k) plan than to pay off the mortgage faster than you need to. This is especially true in those places where interest payments on homes are tax-deductible. You pay them more, but you make the money back - and more - by the investments that you couldn't have made if you paid them back earlier.
You may not have realized, but this is what banks are doing to you when you open a current account - they're taking the money you put into it and investing it in someone who will pay them more interest to hold it for a while (such as a private business taking out a loan). Beat them at their own game! Maybe you got an interest-free overdraft as a special student deal. There's very little reason not to just put that overdraft straight into a savings account and make money off the bank. Don't feel sorry for them - they're too busy making money off the people who just leave all their savings in a current account to care. :-)