Murmansk Oblast is the northernmost region of Northwestern Russia, on the Kola Peninsula, which borders Finland's Lapland to the west, Norway's Finnmark Region to the northwest, the Barents Sea to the north, the White Sea to the southeast, and Karelia to the south.
Murmansk Oblast is of interest to a tourist mostly for its novelty value—it is located in Russia's extreme north, above the Arctic Circle, and is relatively easy to get to from Saint Petersburg. The highlights of a Murmansk Oblast trip are certainly the capital itself and perhaps also an adventurous trip out to Kola. To heighten the novelty value of your trip, try to come around the summer Equinox, when the sun moves around the sky but never sets, or the winter Solstice, when the sun never rises and the city is plunged into blackness. The winter is cold, but it is also the best time to see the Northern Lights.
See Russian phrasebook.
There is also a long (about 28 hour) daily overnight train to Murmansk from Saint Petersburg, which leaves Piter around 18:00 and arrives the next day around 22:00. Trains also travel between Murmansk and Petrozavodsk.
People from Baltic and Scandinavian States, as well as Germany, Poland, Czech Republic, Slovakia, Austria, Hungary, Slovenia and Northern parts of Croatia or Serbia can travel by car. The journey should be carefully planned and can take about 2-4 days (depending on location) one way.
If you travel from Croatia, Bosnia or Serbia you can travel by bus to Stockholm and then travel by train to Kiruna and then to Kirkenes and the border. Such travel can be tiring and good for people with a good health. Oslo airport has a good connections with Croatian, Bosnian, Montenegrin, Albanian or Kosovo cities (operated by Norwegian Air Shuttle and SAS, mostly seasonal). Then you can fly to Kirkenes (operated by Norwegian Air Shuttle and SAS) and then go to the border. If you travel from Serbia, there're year round connections between Belgrade and Stockholm (operated by Air Serbia), then fly to Kiruna (operated by SAS) and then travel by train/bus to Kirkenes and then to the border.