Like the rest of the Kingdom of Denmark, the official currency is the Danish krone DKK. In tourist heavy areas, other currencies such as Icelandic króna, euros, pounds sterling and US dollars may also be accepted but always ask first.
Greenland is still largely a cash economy. With improvements to the infrastructure over the past few decades, the number of merchants accepting credit or debit cards are steadily growing, although many still do not. As a general rule, unless you're dealing with hotels or mainland chains with a presence on the island e.g. supermarkets, don't automatically expect that credit cards are accepted - carry some cash as a backup. Every settlement has at least one ATM and if all else fails banks may be able to give you a cash advance from your credit card.
Tourists to Greenland sometimes buy:
Most flights land at kangerlussuaq, one of those lovely places on earth where you can buy duty-free after landing. stock up on cheap booze, smokes and everything else at prices far lower than the rest of greenland. important: greenland is not a member of the eu, so although you may be travelling from denmark, the custom rules are the same as for a trip out of the eu.
These are the names to look for, if you need to buy groceries:
Pilersuisoq - Chain of larger supermarket usually found in small villages. Has a little bit of everything.
Pisiffik - Chain of larger supermarkets present in the cities.
Spar - Dutch supermarket chain with a few shops in Greenland.
Brugsen - Danish supermarket chain with a few shops in Greenland.