It is fairly universal across the legends constituting the Matter of Britain that Arthur, after he became king, fought in a war of consolidation against the northern British kings and then fought a war against the Saxons. The former is fairly well developed, but the latter much less so. The primary text on the Saxon Wars is that of Nennius, in his Historia Brittonum (sect. 50):
Then it was, that the magnanimous Arthur, with all the kings and military force of Britain, fought against the Saxons. And though there were many more noble than himself, yet he was twelve times chosen their commander, and was as often conqueror. The first battle in which he was engaged, was at the mouth of the river Gleni. The second, third, fourth, and fifth, were on another river, by the Britons called Duglas, in the region Linuis. The sixth, on the river Bassas. The seventh in the wood Celidon, which the Britons call Cat Coit Celidon. The eighth was near Gurnion castle, where Arthur bore the image of the Holy Virgin, mother of God, upon his shoulders, and through the power of our Lord Jesus Christ, and the holy Mary, put the Saxons to flight, and pursued them the whole day with great slaughter. The ninth was at the City of Legion, which is called Cair Lion. The tenth was on the banks of the river Trat Treuroit. The eleventh was on the mountain Breguoin, which we call Cat Bregion. The twelfth was a most severe contest, when Arthur penetrated to the hill of Badon. In this engagement, nine hundred and forty fell by his hand alone, no one but the Lord affording him assistance. In all these engagements the Britons were successful. For no strength can avail against the will of the Almighty.
This seems explicit enough, but runs into the problem that we don't actually know where any of these places are, and the only battle that has any date in any sources is Badon (516 in the Welsh Annals, although Gildas seems to suggest it was before 500). It's also worth noting that (1) Nennius doesn't actually say that these were all in a single campaign, and (2) it's not entirely clear that what the principle of his ordering is. Nonetheless, it's common to assume that this list is at least more-or-less chronological order in a single major campaign. If we do assume that, it's still unclear whether the campaign would have been in the north (Scottish border), or the center (the area around York and Lincoln) or the south (around Caerleon).
Suggestions are plagued by loose fantasies with names, so that pretty much every river in Britain with a name like Glein, Gleni, or Glen, gets to claim the first battle, and so on with the rest. Since many of the names are not distinctive -- Glen would just mean 'pure' and Douglas would just mean 'black water' --this doesn't help much. But on the other hand, it's true that we have very little more than names and basic descriptions. We have a fairly good mix of geographical identifiers, and he order of the battles in terms of geography is: river, river, river, river, river, river, wood, castle, city, river, mountain/hill, mountain/hill. But these are all so generic that they don't provide much guidance. The name that seems most promising is Caerleon (City of Legions); Caerleon in these contexts usually means Caerleon-on-Usk, in the south of Wales, which seems to suggest a southern campaign, and this is a common proposal. On the other hand, "the wood Celidon" seems to some to suggest the Caledonian Forest, which would put it in the north. The primary virtues of a more central campaign are the importance of York (Eboracum) and the ability of the central campaign to allow some northern or southern action if you like.
I suspect the southern campaign option is the most popular one today; one reason for that is historical -- it just does not seem historically likely that there were enough Saxons in the north to be that much of a threat. Nonetheless, the legendarium seems clearly to envisage a northern campaign. In the Vulgate Cycle, the main Old French sequence of stories, the northern kings fail to defeat Arthur in part because their war with him is interrupted by a Saxon invasion of their countries, thus giving them a more immediate problem; Arthur's wars with the Saxons grow causally out of this, so it seems he has to be fighting in the north. Wace seems to envisage a war spanning an impossibly large portion of the island: major waypoints are 'beyond York', York, Lincoln, Bath, and Totnes, but strikingly the whole thing ends at Loch Lomond, with Arthur having invaded Scotland to terrorize the Scots for supposedly having aided the Saxons.
If I want to fill out some of the framework here, I need to make some choice about where the battles would have taken place. The legends seem to push me to a northern campaign. My current tentative idea is something like this:
(1) The river Gleni: Sometimes 'Glein'. The mouth of the River Glen, overlooked by Yeavering Bell (Ad Gefrin, where there is an Iron Age hillfort)
(2), (3), (4), (5) The river Duglas in the region Linuis: Various locations around Douglas Water in Lanarkshire
(6) The river Bassas: This is generally recognized as the most elusive site, regardless of the assumptions one makes, and it's one on which I waver. I'm inclined to make it a minor tributary of the Clyde somewhere in the very broad vicinity of Fallburn Hill Fort or Crawford Castle. A suggestion that is sometimes made is that it is Cambuslang, up near Glasgow, which has some Arthurian associations (although perhaps not close enough to make it stand out) and a nearby Iron Age hill fort. It's also not a stretch for it to be a battle after some battles along the River Douglas. Thus it has some attractions; but then 'the river Bassas' would just be the river Clyde, and I am not sure why it wouldn't just have been called that, since it's one of the rivers with the most durable names.
(7) The wood Celidon: The Great Wood of Caledonia, perhaps near Drumelzier.
(8) Near Gurnion Castle: Near a Roman fort somewhere around Stow of Wedale.
(9) The City of Legion, which is called Cair Lion: I am strongly inclined to go well out on a limb with Carlisle, previously Luguvalium, in Cumbria. It would have been Caer Luel, not Caer Leon. But if Nennius saw anything described as 'the City of Legion', he may have just assumed that it was Caerleon, because this is what 'Caerleon' was taken to mean. York (Eboracum) would probably be safer -- it may have occasionally been called City of Legions, and comes into the campaign anyway, and the legendarium has the Saxons besieging York in the time of Uther. But there was a Roman legion stationed at Luguvalium, and a Roman fort (where Carlisle Castle is currently found); and Carlisle would have been in Rheged, one of the northern kingdoms that the legends suggest were invaded by the Saxons. Some people suggest the Roman hillfort of Trimontium, and it would have been a thriving place, although as far as I know, no one ever calls it 'City of Legion'.
(10) The banks of the river Trat Treuroit: Sometimes 'Tribuit' or 'Trevoit'. Hexham/Warden, where the North Tyne and the South Tyne join to become the Tyne. If one were to do Trimontium, the River Teviot would be the obvious choice.
(11) The mountain Breguoin: Sometimes 'Agned'. The Roman fort of Bremenium, near Rochester.
(12) The hill of Badon: Cockleroy Hill, near Linlithgow. Then Arthur and Hoel could march to Loch Lomond, as in Wace.
In Google Maps, the campaign would look something like this. Of course, many of the particular places are only approximate, and armies would not have been following modern roads except perhaps occasionally where the modern roads happen to follow much older Roman roads. Some of these overlap with places suggested by those who posit Arthur as 'really' a Scottish king; this is not consistent with the legends, but many of the places would make some kind of sense of the legendary battles. You might also notice the regular associations with Iron Age and Roman hillforts. (This would make sense of Nennius's list, I think; it might well be a list of hillforts of which he or his source knew, which then were associated with Arthur. And there does seem a sporadic impulse in the legendary traditions to associate Arthur with Roman and Iron Age hillforts, in much the way that Merlin is often associated with Neolithic monuments.)
