"Viking" swords averaged about 1.2kg. These were one-handed swords, usually used with a shield, optimised for cutting. The typical blade length was about 70-80cm. There was a lot of variation about that average weight, with some surviving examples being below 800g, and the heaviest survivors are 1.8kg (2 examples) and 1.9kg (which is monstrously heavy for a one-handed sword). "Viking sword" is a common name for these, since many examples were actual Viking swords. However, it was a common European style of sword, any many were never in Viking hands. "Viking-Age sword" is more accurate, or perhaps "Carolingian sword", since many were made in Carolingian Francia.

Earlier early Medieval and late-Antiquity swords from Europe and the Western Steppe appear to have often been lighter, with about 600-900g being common for "full-length" swords. On average, swords became longer, and hilts became heavier, as heavy pommels and guards came into use.

Later European Medieval swords averaged about 1.1kg, if one-handed and double-edged. Single-edged one-handed swords (falchions and messers) appear to have been a little lighter on average, perhaps about 900g, but these were often somewhat shorter. Two-handed late Medieval European swords (i.e., longswords, bastard swords, hand and a half swords, rather than post-Medieval big two-handers) averaged about 1.6-1.7kg. See https://www.reddit.com/r/SWORDS/comments/5aoja8/katana_and_longsword_weights_and_lengths/ for some data on such swords.

For non-European swords, I've seen far fewer weights. Near Eastern Crusades period swords (typically broad-bladed double-edged straight swords) appear to have been similar in weight to European swords. One huge example, 90cm blade and over 6cm wide, was 1.28kg, and shorter and narrower examples would probably be lighter. Perhaps about 900g might be typical for one-handed Chinese swords of the time, with a shorter average length than European swords, and about 1.5kg for hand and a half swords, and 2-3kg for huge two-handed swords (which is typical for post-medieval European huge two-handed swords).

Moving to places without swords during the Medieval period (e.g., the Americas, Polynesia, Melanesia, Australia), the closest equivalents to swords were clubs. One handed "full length" clubs averaged about 900g, and two-handed clubs about 1.6kg.