The Historical Legacy Of The Norwegian Vikings
The Norwegian Vikings were by far the most skilled and resilient navigators of all the Vikings during the renowned Viking Age (a sub-epoch or sub-era of the lengthier Early Middle Ages), travelling far and wide distances in their imposing longships over the North Atlantic Ocean on perilous yet legendary voyages of exploration, trade, and conquest which ultimately ended in permanent settlement in some regions with some of the harshest climates on Earth as well. Their impressive historical legacy is recounted in the Icelandic sagas and is represented by the rich cultural heritage that they left behind in their overseas colonies, most notably in the contemporary Nordic nations of Iceland and the Faroe Islands (an autonomous territory of the Danish Realm/Kingdom of Denmark) as well as, historically, in Greenland and Vinland (present-day Newfoundland, Newfoundland and Labrador, Canada). As such, the Icelanders and Faroese are Norwegian by paternal descent as well as Gaelic/Celtic by maternal descent (according to genetic studies), the Norwegian Vikings having previously brought along with them native thralls or slaves as well as nobility from both Ireland and Scotland.
In the British Isles, the Norwegian Vikings (and the Danish Vikings as well) fused their cultural heritage with that of the insular Celts to create a hybrid culture known as Norse-Gaelic (represented by the Norse-Gaels). In Ireland in particular, the Norwegian Vikings established the first towns/cities, namely Dublin (once known as Dyflin in Old Norse, meaning black pool, and initially founded as a longphort), Cork, Limerick, Waterford, or Wexford (while also establishing the first trade routes on the island). The Norwegian Vikings also played a very important historical role in the early medieval history of the Isle of Man.
Regarding the Faroe Islands and Iceland, it must also be mentioned that both the Icelandic language and the Faroese language are the closest living North Germanic languages to Old Norse (from which they descended, alongside all other North Germanic languages plus the extinct Norn language, spoken a long time ago in the Orkneys and Shetland) as well as to several dialects of the Norwegian language from south-western Norway.

Longships with Norwegian Vikings setting sail from Sognefjord, Vestland fylke, western Norway, the longest and deepest fjord in Norway (as well as the second longest fjord in the world after Scoresby Sund from the eastern coast of Greenland which faces the Greenland Sea), as imagined by Norwegian romanticist painter Hans Gude. Image source: Wikimedia Commons
It is well known from medieval sources that the Norwegian Vikings generally sailed westward (in stark contrast to the Swedish Vikings who sailed across the rivers of mainland Europe, voyaging deep inland within Eastern Europe and venturing as far as east as the Caspian Sea in Asia). At the same time, compared with the Danish Vikings (who also set sail westward to Anglo-Saxon England and, to a lesser extent, also to Ireland) the Norwegian Vikings pushed the limits of European navigation and exploration beyond the confines of the North Sea. So it is that they went far beyond the British archipelago, passing by both the Albion (i.e. Britain) and Hibernia (i.e. Ireland) in their relentless search of new lands to explore, colonise, herd, and farm.
As previously mentioned, these were, in chronological order, the Faroe Islands, Iceland, Greenland, and Vinland in North America. As an important side note, with respect to the Faroe Islands, it must be mentioned that this relatively small archipelago acted as a transport hub for the Norwegian Vikings approximately halfway on route from Norway to Iceland where they were able to rest and stockpile resources throughout their long voyages. Moving forward to North America (after discovering Helluland and Markland), in Vinland more specifically, because of the skirmishes the Norse had with the Native American population whom they called the ‘Skrælings‘, they eventually had to retreat and leave Newfoundland.

A general contextual territorial map of Northern and Northwestern Europe during the Viking Age (as part of the Early Middle Ages). Image source: personal annotated creation based on a map by NASA (more specifically NASA Landsat imagery) released in the public domain via Wikimedia Commons
Therefore, in navigational regards, the historical legacy of the Norwegian Vikings is represented by the long and harsh maritime routes that they voyaged over the harsh and (sometimes) treacherous waters of the North Atlantic Ocean in search of lands previously unbeknown to many other Europeans of their times, exploring these regions, settling these new territories, and connecting them through a durable trade network in a vast colonial empire situated in the Northern Hemisphere which also encompassed several vassal states (e.g. the Kingdom of Mann and the Isles). Last but not least, it is not that far fetched to believe that the Norwegian Vikings also sailed to and reached the Svalbard Islands (although no solid archaeological proof has been discovered to date in this particular regard), once again pushing forward European medieval navigation at that time (going well beyond the Norwegian Sea to the north in the process, naturally).

Amateur map annotated by me depicting the voyages of the Vikings based on a map published using NASA World Wind which I found on Wikimedia Commons. Image source: Wikimedia Commons
Documentation sources and external links:
- Personal knowledge and personal historical research
- Saga literature (in this regard, if you are interested to know more regarding this subject/topic, please also see the related article 10 Epic Medieval Icelandic Sagas One Should Absolutely Read)
- The Sagas of the Icelanders by Jane Smiley (on Google Books)
- The Viking Hondbók: Eat, Dress, and Fight Like a Warrior by Kjersti Egerdahl (on Google Books)
- The City of Dublin on www.vikingeskibsmuseet.dk (in English)