Most readers will have some preconceived notion about what the the terms Norse and Old Norse mean. Even if your understanding of the subject is vague, you probably know that Old Norse is famously the language of the so-called Vikings – a historically recent ethnic term that is often used interchangeably with Norse. In casual conversation this works fine, and you're not confusing anybody by using any of the two. Academically (of course) it's much more complicated. If you look in an English dictionary dating to, say, the early 1900's you are likely to see Norse defined simply as «Norwegian». Historically this is what the word meant, but the meaning has since shifted. Today it generally refers to the Germanic speaking Scandinavian population, as well as that of their overseas colonies during the Viking Era and Middle Ages. As for the term «Viking» denoting Norse peoples: This is somewhat misleading because a viking was actually a specific kind of person within Norse culture. A title, really. But owing to its prevalence, scholars frequently resort to it for convenience. For example, Norse overseas colonies in places like Ireland, Greenland, and Russia, is sometimes referred to as The Viking Diaspora. But that's a discussion for a later time. Let's stick with Norse for now.
The first Norsemen
Outside the scholarly world, the term Norse is most commonly applied to the Viking Era, which kicked off in the 8th century for reasons varied and unknown (innovations in sailing technology is commonly assumed to be one factor). The Old Norse language was more or less fully developed by the 700's, having sprung from the preceding Proto-Norse or Proto-Scandinavian tongue (pick whatever term suits you, there's no consensus!), which in turn developed from a Northern dialect of Proto-Germanic.
For Norway, Greenland, the Faeroe Islands, and Iceland, it makes sense to talk about common Old Norse all the way through the 14th century, when the Black Death unleashed a period of Mad Max-esque socioeconomic and demographic turmoil, forcing Norway into a linguistic revolution that gave birth to the Middle Norwegian language. This made mutual intelligibility with Iceland a thing of the past.
Middle Norwegian is basically the sort of grizzly post-apocalyptic linguistic change you can expect when something like three fifths of the entire population shuffles off their mortal coil, leaving their brats with nobody to correct their baby language and youthful slang. Up until then, the differences between Norwegian, Faroese and Icelandic had been a matter of dialect. Specifically the Old West Norwegian dialects of the early settlers.
It wasn't a unified language
Anglophone scholars divide Old Norse language into West Norse, which I described above, and East Norse which encompasses Old Swedish, Danish, and Gutnish (spoken on Gotland). These are distinct enough to be considered languages of their own. Still, all these variants of Old Norse were mutually intelligible to the point where West Norse speakers accepted the term Danish Tongue (dǫnsk tunga) as a name for their own language. This likely originated among Anglo-Saxons to describe the language of Scandinavian settlers, traders and raiders, drawing a generalization from what seemed to them the dominant culture – namely the Danes. This isn't entirely dissimilar to how we generalize by calling them Vikings today. This common linguistic identity, and the fact that they adopted a foreign term as their own, seems to suggest a sense of cultural kinship among Viking Era Scandinavians. Swedes, Danes and Norwegians can still understand each other with relative ease, which is a fantastic linguistic privilege. And still, when reading Norwegian, Danish or Swedish runic inscriptions from the Viking Era, we may enjoy the distinct and recognizable traits of each.
Terms, translations, and turmoil
Despite mutual intelligibility and common heritage, Nordic scholars are not on the same page as English ones when they discuss Old Norse in their native languages. Scandinavian scholars use the term «norrønt» which overlaps with Norse. But confusingly, it does not mean exactly the same. Firstly, the term excludes Swedes and Danes, and refers exclusively to the West Norse language(s) and populations of the Viking Era and High Middle Ages (8th through 14th centuries). One can easily write entire books about why this is, as people have, but bear with me.
One obvious reason is the historical and linguistic divide between Sweden and Denmark on one side, and the entire Western Nordic world on the other. Swedish and Danish language rapidly developed away from Old Norse in the High Middle Ages: 13th century Old Danish looks a lot more like modern Danish than 13th century Old Norwegian resembles modern Norwegian. In fact, most Scandinavians would have an easier time reading 13th century Danish, than they would trying to make sense of Old Norse. This means that Old Norse is a viable term for language in Norway and the Western Nordic from the 8th through 14th centuries, while 13th and 14thcentury Swedes and Danes spoke a different language entirely, though Norwegian would soon enough make a similar turn.
The Swedish dialect of East Norse, so-called Runic Swedish, appears before 800 and is gone by 1225, superseded by Old Swedish until 1526. The heavy West Norse connotations of Old Norse and «norrønt» also rest on the fact that the vast majority of surviving Norse literature comes from the Western Nordic area: Iceland, and to a far lesser extent Norway. Norway tends to mooch off Iceland, because Icelandic identity was reflexive towards Norway. They wrote a ton of historical fiction placed in Norway, and composed some great propaganda pieces for the Norwegian crown. Besides, Icelanders pretty much considered Norway the womb of the Icelandic nation, and suffered immense Norwegian cultural and political pressure.
Very few vernacular manuscripts remain from the other Nordic countries, but the literature and implications of identity permeates all Nordic scholarship on the matter to some lesser or greater extent. I will not go into the long-term political history of Scandinavia and the Nordic, but they all have in common a heritage of Norse prehistory that functions as a serving bowl of national myths of origin, which is politicized accordingly.
Norse culture and contemporary identities
The reception of Old Norse, viking, and medieval history is treated variously between the Nordic countries. Denmark and Iceland, it seems to me, are the best when it comes to popular representation of the Viking Era in particular. Denmark is rich in physical remains but short on written sources, with Iceland it's by far the other way around. And each country infuses the era with their own particular brands of plushy patriotic sentiments. Warning – stereotypes ahead. In Norway, as well as in Norse texts, the term Norse (Norwegian: norrønt. Icelandic: norræn) have strong norwegiocentric connotations. I've told Icelandic barhoppers I do Old Norse and they've corrected me to Old Icelandic, while Swedes might have no idea what I'm talking about when presented with the term.
Such identity markers are visible even in the world of Academia, down to the spelling you see in the critical editions of Norse texts at your university library. In Iceland, as with the majority of international scholarship, Old Norse is taught with modern Icelandic orthography and pronunciation. This is what you hear in virtually every bedroom video guide to Old Norse pronunciation, which are a dime a dozen on YouTube. Norway is the only country (as far as I'm aware) where reconstructed pronunciation is taught. Orthography follows a convention called «Classical Old Norse», which intends to portray a normalized version of 13th century Norse spelling. This is the tradition I was brought into, though it appears to be rapidly disappearing. To this day, years after my indoctrination graduation, I still get A Clockwork Orange-like fits of nausea every time I hear Old Norse pronounced as if it were modern Icelandic. I'm sure the feeling is mutual.