I have noticed that in the later Roman period the Scutum seems to have been replaced by a more rounded shield and that 'barbarians' had very similar shields to that of a Viking. How did the Scutum stop being the standard shield of the Roman army?
While the Scutum is a popular depiction in several pieces of monumental art, unfortunately, we really don't know if it was actually a standardized shield of the Roman legions, and if it was, for how long. Unfortunately, we only have one surviving example of the shield design itself.
As for possible reasons why it was phased out, if we assume that it was standardized? Cost and complexity is certainly one possibility. While the square shape of the shield is not all that complex, the curve is definitely more complex than a rounder tall shield. Similar to the phasing out of the lorica segmentata, this shield may have been phased out in favour of equipment that is easier to mass produce. There is also the possibility that the shift in tactics of the Roman legions during the later imperial period played a role as well. Roman infantry shifted from being primarily heavy infantry armed with short swords for thrusting to progressively longer swords, and eventually spears. These longer weapons required more space with which to maneuver, and the scutum is particularly ill-suited to spear men as compared to a rounder shield.
There is no evidence of the scutum ever becoming the standard shield. For example, the Praetorians always had a rounded shield whereas description of legionnaires varied. The surviving example of the Teutoburg Forest battle (together with the surviving Lorica segementata from the same battle) suggests that there was no standard armour ever and that equipment was used due to personal preference and availability. Thus, it is quite likely to see mail-wearing soldiers along segmentata and hamata wearing soldiers, with all of those using a combination of shields.
All descriptions of Roman battle tactics show that despite the change in shields (if there was one) and swords (longer, possibly due to a need of penetrating more armoured enemies) roman battle tactics essentially remained the same in close combat throughout the whole empire (for example, Ammianus Marcellinus description of sword combat in the fourth century AD is nearly impossible to distinguish from sword combat of the 1st century BC).
So the questions we have to ask ourselves are:
Was the scutum actually a better shield? This one is hard to answer because while it is slightly larger, that does not automatically mean better. Weight is a big issue and we don't really know if the rounded shields were inferior or superior levels of protection.
Is there a tactic/formation that is impossible to do with the long rounded shield that is possible with the scutum? I can't think of any. The testudo was able to be performed with the rounded shields.
If we assume there was a change in equipment over the later centuries one possible solution is that with the added weight of the missile weapons (missile weaponry increased in number throughout the centuries) and defensive equipment (depending on who you believe helmets may have become heavier and there might have been there was more armour in general in response to the now heavier-armoured forces of the enemies of Rome) soldiers might have looked for weight savings.
Again, this is conjecture though because we don't really know if weight savings in the marginally smaller surface area ever really amounted to much or if they were just eaten up by heavier shield bosses or more metal straps to on the shield to improve durability.
Another problem on the monumental evidence is that sometimes we don't know if the forces depicted there are dismounted cavalry or infantry (obviously this would make a real different interpretation of shield types, especially of the more rounder version of Roman shields). This is for example true of the arch of Galerius.
A point on barbarian shields - we don't really know how much representative those types are as well, as we don't really have enough surviving examples or pieces to make any kind of statistically representative collection of evidence.
TL, DR: Not enough evidence to really come to any conclusions either way.
EDIT: If you are interested in the current debate I suggest you pick up the great "War and Warfare in Late Antiquity. Current Perspective, Alexander Sarantis and Neil Christie (Editors), Leiden-Boston 2013), which covers a lot of ground and is the best current work on the subject.