The two-handed sword (the proper claymore or big sword - also known as a two-handed sword translated from Gaelic) was slung on the back. Being a big double-handed beast, it was more likely to be used by a big-built warrior. It was basically the Celtic version of double-handed swords being used in the late middle-ages/renaissance in Western Europe. In the early 16th Century, German Landsnechte employed big, strong "crazies" armed with double-handed swords who went in front of their columns of pikemen and scythed a path through the enemy's forces while chanting along the lines of "Here we come !" Highlanders adopted weapons through contact with other armies, but kept them long after others superseded theirs.
They were carried as late as the 1745 Jacobite Rebellion, whilst during the first rebellion in 1689, claymores did terrible damage at the Battle of Killiekrankie and eye-reports talked about the numbers of severed and cleft heads and severed limbs found on the battlefield.
Basket-hilted broadswords made their appearance in the Highlands in the 17th Century, most likely as the result of contact between Highland mercenaries and continental dragoons during the 30 years' war who carried basket hilted, straight bladed swords. During the War of the Three Kingdoms (aka English Civil War), such swords (called mortuary swords) were obtained and also copied by Highland and Scottish smiths. Double-edged, the most prized had Spanish blades made by Andreas Ferrara, a mark of quality sought after by Highland gentlemen and chieftains, right until weapons were banned in 1747. A variation on the broadsword, with a curved blade appeared around the end of the 17th Century. A lighter variety of broadsword, with a narrower blade sharpened along one edge only, called a basket-hilted backsword became popular in the early 18th Century and was adopted by the Highland regiments of the British Army (as used today).
Other weapons adopted by Highlanders during the middle ages included the targe/target (copied and developed from small round shields carried by Scots and other armies' infantry) and the dirk (ditto where it was called a bollock dagger).