Getting in is expensive and time-consuming. In legal theory, citizens of the 41 signatories of the Svalbard treaty including such unlikely countries as Afghanistan and the Dominican Republic need no visas or other permits to visit – or even work – in Svalbard. However, in practice it's difficult to arrive in Svalbard without transiting through Norway, and as Norway considers Svalbard a domestic destination, you'll need to pass through Norwegian immigration first. In the other direction, Norway reserves the right to check the passports of passengers coming from Svalbard.
By plane
Longyearbyen has the largest airport on the islands : LYR (http://www.avinor.no/en/airport/svalbard/frontpage). SAS Scandinavian Airline Systems (http://www.flysas.com/) has scheduled flights from Oslo IATA: OSL (http://www.osl.no/) Su-F 4.5 hours, US$150-350 each way, and from Tromsø IATA: TOS (http://www.avinor.no/en/airport/tromso) Su-Th 1.5 hours, US$100-300 each way.
There are also occasional charters from Murmansk IATA: MMK (http://avia.murman.ru/) or Moscow.
SAS considers flights to Longyearbyen from Oslo or Tromsø domestic, so a SAS EuroBonus award ticket from anywhere in Scandinavia to Svalbard costs just 12,000 EuroBonus points. This little loophole is well known by SAS frequent flyers and award availability is quite limited, so book well in advance if planning to use this. SAS flights can also be bought over the Internet either direct from SAS or via certain meta agents. The majority of flights are very expensive when booked on line but if you book well in advance you can usually find a few cheap flights a week.
If you are flying to Longyearbyen from far away, a cheaper option may be to book a round the world flight RTW. RTWs are specially-priced tickets which cover travel over several continents, several countries, or a certain large number of miles.
Note that some flights arrive at odd hours such as 3AM. Many outdoor activities on Svalbard demand that you are 100% fit, and you might not be if your flight arrive at 3AM and your activity starts at 7AM.