Vercingetorix, last leader of the Celtae or 'Gauls' surrendering to Julius Caesar after his defeat at Alesia
The issue of who the Celts were and what role they played in Scottish history (and for that matter, British, Irish and European history as a whole) is a topic which has been much discussed among historians and archaeologists, especially over the last few decades. In the popular consciousness, the term 'Celt' is considered synonymous with those peoples who historically spoke languages that were noticed to have been closely related; including the Gaelic languages of Ireland, Scotland and the Isle of man and the Brythonic languages of Wales, Cornwall and Brittany. Therefore, anyone with heritage from any of these countries can potentially claim to be a 'Celt' and see this as part of the shared kinship between these countries (in contrast to the Germanic England and her folk, for instance).
The term is ethnically and potentially politically loaded in this way, as far as it points to a shared kinship not just between the pre-Germanic peoples of the British Isles, but also with the ancestors of the French and Spanish; who now speak languages derived from Latin, but who once spoke tongues akin to those that could be heard across the British Isles before the arrival of the Anglo-Saxons and Norsemen. Yet, the term has come under fire from academics who have noticed that historical sources never apply the term 'Celt' to anyone living in the British Isles, and that the word was first coined in 1707 by Edward Lhuyd, a Welsh linguist who noticed the similarities between the aforementioned languages, as a catch-all term.
For this reason, according to some academics, we are better off discarding the notion of any 'Celts' being associated with the British Isles. However, with up-to-date genetic information, we are now in a better position to make definitive statements (at least in comparison to prior decades) about who the Celts were and how exactly the term can be applied. Much of what we understand about 'Celts' comes from how historians of the 18th to early 20th centuries understood prehistory, which in itself was largely based on historical sources from the Graeco-Roman world (which is understandable, as archaeology was in its infancy throughout the 18th and 19th centuries).
The first issue of who the Celts were has to do with the origin of the term itself, which is the Latin word Celtae, which Caesar claimed a people who lived in modern-day France (now known to us as 'Gauls') called themselves (the Romans called them Galli which means 'chicken' in Latin, an interesting point when one considers the chicken is still the national animal of France). However, the word Keltoi in Greek was also known and was used to describe a horde of barbarians from the north-west who ravaged Greece and the Balkans; some of whom even settled in Turkey.
Therefore, for early archaeologists wishing to discover more about the Celts, it was then their task to find evidence of material culture predating the Greeks and Romans in Western Europe. In 1857, drought affecting Lake Neuchâtel in Switzerland led to the exposure of many prehistoric artefacts, particularly near the town of La Tène. Swords, scabbards, spearheads, shields, axes, knives, horse harnesses, rings and brooches were all found to have been deposited in the lake, the context implying this was done as part of religious ceremonies.
An example of a La Tène style horse harness fitting from Britain, note the swirling patterns and enamelling
However, the key element is that objects of the same style have been found all over Europe corresponding to areas where historical Celts were known (including Turkey). Therefore, archaeologists began to theorize that the 'La Tène culture' was spread across Europe by those marauding Celts mentioned in Greek and Roman historical sources beginning c.450 BC. Additionally, antecedents to La Tène could be found within another culture named after the village of Hallstatt in Austria, where extensive salt-mines have been found and are associated with a rich material culture centered in the Alps, but which too can be found across Western Europe beginning c.800 BC.
The lakeside town of Hallstatt, Austria, which was once home to a salt mining complex
On this basis, the Celts have traditionally been associated with the Hallstatt and La Tène cultures specifically, and so they were believed to have been responsible for the spread of the 'Celtic' languages across Europe. However, particularly since the 1960s, widespread questioning of assumptions about migration being the primary vector of linguistic and cultural spread came into vogue among academics. There are certain instances in which this skepticism is indeed warranted: for example, Celtic languages are well-attested from inscriptions in parts of Spain and Portugal, but La Tène culture in particular is practically absent from those countries. Therefore, if the Celtic languages were spread by bearers of that culture, surely we would expect to see more of it in the Iberian archaeological record than we do?
At the same time, we now know that such assertions as that there were 'no Celts in Britain' is a step too far, as very recent genetic evidence has emerged to show us that there was migration to Britain which appears to have been responsible for bringing both the Hallstatt and La Tène cultures. A 2021 study by Nick Patterson et al. entitled Large-scale migration into Britain during the Middle to Late Bronze Age contains samples from both the Bronze and Iron Ages, and while the authors focus on overall genetic admixture to argue for continual migration throughout the Bronze Age, the male haplogroup samples from the Iron Age suggest input at that time too.
Specifically, the haplogroups known as R1b-U152 and G2a-L140 (more common in France than Britain) don't seem to have been present in Britain before the Iron Age, while R1b-DF27 ('brother' of U152) was a small minority in the Early Bronze Age, but had increased in numbers by the Iron Age (it seems to have come from Spain and could be the origin of Irish claims to Spanish origins. The majority of men in both Bronze and Iron Age Britain belonged to R1b-L21, which is now still the most common haplogroup in Scotland, Ireland, the Isle of Man, Wales and Brittany. R1b-L21 first arrived in the British Isles at the end of the Neolithic with the Bell Beaker culture, so while we can safely dismiss the idea of Iron Age Celts replacing the local population in any part of Britain, it seems clear that there was migration and that this can be associated with the introduction of the Hallstatt and La Tène cultures.
Haplogroup R1b-DF13, largest branch of L21, was once the most common haplogroup in Britain before the arrival of the Anglo-Saxons
I would also argue that Celts arrived in Ireland and Scotland, as while there is no evidence of any of these Continental haplogroups from Iron Age Ireland so far, many men who claim descent from the King of Leinster known as 'Breassal Breac' (who reigned some time between the 5th and 3rd centuries BC) carry the same subclade of R1b-DF27 as men from Iron Age England (known as 'Z31544'). This means that we have certain evidence that Celts were settling in Southern and Eastern England, as well as possible evidence for their settlement in South-East Ireland, during the Iron Age. Similarly, another subclade of DF27 (L165) is most common in North-West Scotland, particularly among members of Clan McLeod.
At the same time, the authors of the aforementioned 2021 study (though it only dealt with Britain) concluded that migration was continually flowing to England and Wales, but not to Scotland, during the Bronze Age. The Iron Age samples from Scotland also show no evidence of incoming haplogroups that were not already present during the Bronze Age. This means two things: firstly, we can confidently say that if the Celts were coming to Scotland (assuming that R1b-L165 represents them), they were arriving in very small numbers compared to Southern and Eastern England in the Iron Age. Secondly, this makes it unlikely that the Pictish or Cumbric languages (which are known to have been Brythonic Celtic languages closely related to Welsh) were introduced to Scotland by the Celts.
Therefore, it would make sense to look further back to the Bronze Age for the origin of the Celtic languages. Ian Armit in his book Celtic Scotland (1997, most recent edition 2016) argues that the Celtic languages developed through trade networks across Western Europe during the Late Bronze Age. This makes the most sense to me, as the archaeological evidence suggests that Scotland was much more interconnected with her neighbours across Europe at that time, before the climate took a significant downturn (becoming colder and wetter) and she (as with much of the rest of the British Isles outside Southern England) became isolated c.750 BC.
Armit's justification for using the title Celtic Scotland is that it can be strongly inferred that the Iron Age Scots were Celtic-speaking, even if this had nothing to do with the people called 'Celts' by Ancient Greek or Roman historians. The fact seems to be that the local Iron Age Britons and Irish were closely related to the Celts of modern-day France (indeed, haplogroup R1b-L21 falls under the same P312 mutation as the more French U152 and Spanish DF27). They all descended from the Beaker Folk who took over Western Europe at the end of the Neolithic and have their ultimate origin on the Russian Steppe; a fact acknowledged by both the Pictish and Gaelic origin myths which place their roots in Scythia.
Haplogroup R1b-P312, which peaks along the Atlantic coast
In summary, the issue around the term 'Celtic' is similar to 'Germanic'; an entire cultural-linguistic family has been named after one specific nation within each family. Should we consider Scandinavians not to be 'Germanic' because they're not German? Of course not, because they share common genetic and linguistic roots, but have since differentiated into distinct nations. Similarly, there are enough genetic and linguistic similarities between the Brythonic and Gaelic peoples of the British Isles with their Continental cousins to warrant a catch-all term under which to consider all of them; for which 'Celtic' seems perfectly appropriate (and which would probably be too difficult to dislodge in the public perception anyway).