There is an interesting piece a here on new cell-phone technologies, that allow individuals to make small decentralized transactions without currency. I like to tell students stories about why we still need some currency in circulation. These stories often involve small hot-dog vendors who won't take credit cards, but clearly that could become a thing of the past, along with the problem of splitting up the dinner bill among 12 people. Certainly the experience of us relatively wealthy people is that currency is much less useful than in the past - few retailers will turn down a credit or debit card, and even the short trip to the ATM may not be worth the bother. However, if we look at the data, the use of currency in the world is remarkably persistent. I didn't take the effort to give you references to these studies (I'll let you find them yourself), but the numbers I remember are the following (check for accuracy if you want). One hard number is the following (it's in my textbook). The quantity of US currency in circulation in the world was $2776 per US resident in April 2009. That's a lot of currency - the stock held at any point in time is about 6% of US annual GDP. In case you think that's all held overseas, a study by the Bank of Canada (in the Bank of Canada Review - look it up) shows that Canadian currency outstanding is about 3% of annual Canadian GDP, and most of that has to be in Canada. That's still a huge amount of currency, and that quantity does not appear to have been falling recently. In some other countries people hold even more currency, for example the Japanese hold a quantity of currency which is about 15% of annual GDP. Fernando Alvarez (again, look it up) at the University of Chicago documents a persistently large stock of currency held in the world.
How do we reconcile these aggregate observations with what we observe at an individual level? First, it appears that the denomination structure of the currency in circulation has changed, with less low-denomination notes and more high-denomination notes. We all know what that is about. Currency is the principle means of payment for drug dealers and for people who want to evade taxation. Second, survey evidence for the US (e.g. the Federal Reserve's Survey of Consumer Finances) tells us that a significant fraction of the population (mainly in poor urban areas) have no relationship with a bank. These people are executing all their transactions using currency.
What implications does this have for monetary policy? First, currency is still important. After all, in normal times, most of the Fed's liabilities consist of currency - the stuff is financing the central bank's portfolio. We want to understand what motivates people to hold currency, why they want to hold so much of it, and what the prospects are for currency-holding in the future. All these things matter for how monetary policy works. Second, in terms of optimal policy, there may come a day when it seems appropriate that the central bank withdraw all currency from circulation. Though currency is the medium of exchange for the poor, central banks may ultimately decide that currency is not worth the bother. We have to pay the costs of thwarting counterfeiters by designing the currency appropriately, we have to print new currency to replace worn-out notes, and for what? To provide a medium of exchange for criminals?
Now, what happens if we get rid of currency? Does this mean that monetary policy has nothing left to do? Absolutely not. The central bank still has a role as the lender of last resort and in most countries it runs an interbank payments system - Fedwire in the US. The liabilities of the Fed - reserves - are the medium of exchange in the payment system. Outside money (reserves) can finance a central bank portfolio of assets (just as reserves are now financing more than half of the Fed's portfolio), and open market operations in the usual sense can matter, even without currency in circulation.