Money:
The official currency in Vietnam is Dong. The Dong is non-convertible and in summer 2014 trades at approximately 21,000 to USD. You can use the website of Vietcombank to see the daily exchange rate. The U S dollar, preferably crisp clean bills, is widely accepted among major shops and restaurants. The downside to this is that the prices will be converted from Dong at the vendor's chosen exchange rate, which may or may not be close to the official exchange rate, and will be rounded up to the nearest USD, making it more expensive than the cost in Dong.

It is illegal to list prices or ask for payment in any currency that is not the Dong.

The Euro, Pound and various other currencies are also easy to exchange.  If you intend to exchange your home currency for Dong, ensure your notes are clean and undamaged, as banks and other exchanges will not accept any notes which are torn, excessively crumpled or have writing on them.  Visa and Master card are becoming more accepted in many of the bigger hotels and restaurants, especially in the larger cities with usually a surcharge of 3%. 

Make sure that the Vietnamese notes you receive are not torn, this is because many shops and restaurants will not accept them. Also try not to change too much money at one time, as you will end up with a large wad of notes. The largest denomination is currently 500,000 dong (approx. USD 24). Be careful, the 20,000 notes look only slightly different from the 500,000 one but the value is a big gap. Handy tip: Keep 500,000 dong notes separate from your other dong notes. Less chance of confusion then. Other paper notes are 1,000, 2,000, 5,000; 10,000, 20,000, 50,000, 100,000 and 200,000 Dong. Most are clearly identifiable by colour. 

When you are agreeing prices with Taxi drivers or shops, always use Dong to save arguments later about the exchange rate you were expecting.  While most will use the official rate of the nationals banks, some do try to argue differently.  Always double check the conversion rate you have been offered.

Also with such high denominations of note, be careful of common scams run by street sellers where you are short changed by a factor of ten (eg as part of your change, you may receive five 2000 dong notes as "100,000 VND!").

Outside Vietnam the Dong is normally not accepted (excluding Cambodia and Laos), so before leaving the country remember to exchange any Dong left. In Saigon airport you can change before immigration at two bank booth that use the standard rate plus 2% fee. After immigration there are two more small booths that charge no fee, no receipt and standard rate. On arrival there are two exchange booth with a normal exchange rate and a bunch of ATMs outside turning righ.

Banking:
There are a number of international banks operating in both Hanoi and Ho Chi Minh City with 24hour cash withdrawal facilities. Most ATMs enables you to get cash from VISA, MASTER CARD, Cirrus, Maestro, Plus and JCB network.
Under Vietnamese law, ATM may only dispense cash in Vietnamese Dong. Current amounts that can be withdrawn at any one time is usually very low, 2,000,000 Dong with an additional usage charge variable from 30.000d up. On March 2014 the situation is getting worse particularly for debit card using MasterCard circuit (Cirrus, Maestro), while Visa debit is accepted by more ATMs. Banks are increasing fee against foreign debit cards, both Techcombank and Vietinbank charge 55.000d (the last even without showing the amount, just a generic message "There will be fees", or even without a message). Avoid both, use SCB. Vietcom bank usually charges 20,000 dong.

In HCM City the best choise is still (September 2014) Commonwealth Bank (Australia) with zero fee and a max limit of 25.000.000d (max 100 notes, 500.000d), 4 ATMs around Bui Vien St, no plan to extend the net ouside HCMC.
Citibank ATM at HCM Airport permits withdrawals of 8,500,000 Dong. Donga can go up to 5.000.000 Dong but it only accepts card from the Visa circuit, no MasterCard.
There is nothing to stop you putting your card back in to get more money out, but remember you are still subject to your own bank's daily cash withdrawal limits, about 500E for Maestro/MasterCard network in Europe, and you pay your bank fee every time. Also, ensure you tell your bank before travelling to Vietnam if you intend to use your card to withdraw cash from ATM's. The machines here use the magnetic strip not the more secure chip technology.

Note for Australians:   Stored value cards where you preload with one or more foreign currencies are generally poor value.  To make up for "no fees", they generally offer a poor exchange rate.  Getting any unused funds back once you're home may also be difficult.    You can benefit if the exchange rate drops dramatically after you have loaded the card, but it could also go the other way.
Two cards available to Australians are the Citibank Plus visa debit card, which uses your own funds, and the 28Degrees MasterCard, which is a credit card (and which from Jan 1 2014 will charge hefty fees for cash withdrawals even for cards which are in credit). Both have no foreign exchange fees and one of the best exchange rates you'll get.  The 28Degrees card was the Choice Magazine "Travel Card of 2012".

Note for Italy: Only one fee free debit card exist, CheBanca (Maestro/MasterCard), zero fee, zero cost, standard MasterCard exchange rate but like other MC card limited support in Vietnam.