Great question. First off, the three age system of Stone Age, Bronze Age and Iron Age is a classification of prehistory rather than history. It ends with the Iron Age, after which most societies come into the written historical record, and historians use more fine-grained and culturally specific periodisations: late antiquity, early modern period, Ming dynasty and so on. I'm going to assume that's what you're asking about, as opposed to "ages" in historical periods, so apologies if none of this is what you're actually interested in!

But yes, the three age system is something archaeologists almost everywhere (with the notable exception of the western hemisphere) use on a day to day basis. It isn't exactly a three age system any more: the Stone Age invariably gets split up into the Palaeolithic (Old Stone Age), Epipalaeolithic or Mesolithic (Final Old Stone Age/Middle Stone Age) and Neolithic (New Stone Age); and in most parts of the world another period is inserted between the Neolithic and Bronze Age, called the Copper Age, Chalcolithic or Eneolithic. But essentially it's still the primary chronological system of reference for prehistorians.

Which is funny because it's actually a terrible way to talk about time, even as shorthand. To explain why let's backtrack a little. The three age system was devised by the Danish antiquarian Christian Thomsen in the early 19th century, with the simple observation that the prehistoric assemblages (artefacts found together) were dominated by either stone, bronze or iron tools. Thomsen classified the artefacts in his museum based on this and in doing so was the first person to use typological dating, because it was assumed that iron came after bronze came after stone. As crude as it sounds, typology was the only way of dating prehistoric sites and artefacts until the radiocarbon revolution in the late 1950s.

By the early 19th century though archaeologists had moved beyond Thomsen's three age system and were using much more sophisticated methods of typological dating. They looked at things like pottery, which were far more abundant, and rather than material or utilitarian functions they classified them based on stylistic elements. This method identified much finer types which could be dated relative to each other using a method called seriation. People even attempted to put an absolute calendar date on them by linking types in different places to each other and then to the historical record in places like Egypt and Sumer where it went back far enough (as it turned out these dates were spectacularly wrong, but hey that's science). Archaeologists had also noted by this point that stylistic variation in space was just as, if not more, significant than in time, and the idea of tracking prehistoric "cultures" and their movements preoccupied early 19th century archaeologists. So Thomsen's three ages were eclipsed by finer spatio-temporal types defined from detailed excavations and artefact studies at certain major sites, e.g. "Vinča D" or "Geometric Kabaran".

Now that we can date artefacts reliably and absolutely using radiocarbon dating, and more sophisticated ways of characterising stylistic variation (e.g. multivariate statistics), fine typological schemes like these are, thankfully, on the wane. So why does the even cruder three age system persist? We can thank/blame V. Gordon Childe for re-invigorating the concept. Childe was a hugely influential and somewhat maverick prehistorian and while most of his contemporaries were preoccupied with marking out, mapping and chopping up cultures (a concept he himself can be credited with introducing into English-speaking archaeology), he was still thinking about the long term patterns of social evolution that had interested earlier generations. He was also a committed Marxist, and it was a combination of these two streams of thought that he brought to long term periodisation. Like Thomsen, he thought that progression of material culture through different ages was a sign of progress. But while Thomsen only had access to material from Denmark, Childe was a famous globe-trotter, and could see that people in different parts of the world clearly progressed through these stages at different rates. So Childe tried to map archaeological ages onto what 19th century cultural anthropologists described as the three stages through which society evolved – savagery, barbarism and civilisation – and to contemporary anthropological work on modes of subsistence. What he came up with was: Palaeolithic = hunter-gatherers = savagery; Neolithic = village farmers = barbarism; and Bronze/Iron Ages = city-dwellers = civilisation. In between these ages he posited there were revolutions in the Marxist sense – radical breaks in the social order caused by technological progress in the means of production – the Neolithic revolution when people invented farming, and the urban revolution when people started living in cities. So after Childe the three age system took a peculiar form that you might call the +/- three age system. As I mentioned before, you almost never hear the words Stone Age amongst archaeologists, it's always subdivided into Palaeolithic, Neolithic, etc. That's because for Childe the revolution that occurred between the Old Stone Age and the New was the foremost turning point in world history. The transition between the Neolithic and the metal ages also maintains its relevance because it's roughly where we see a jump in social complexity and, in parts of the world, Childe's second, urban revolution. But the Bronze/Iron Age distinction can't be fitted into his scheme of social evolution very easily and consequently it's a much fuzzier and less talked about boundary.

So today the +/- three age system lives on as a peculiar mix of two very unfashionable paradigms: typological dating and social evolutionism. It can be a useful shorthand. Say I meet someone studying an unfamiliar part of the world for me, if they tell me that they're looking at the Neolithic I immediately get a rough idea a timescale, the type of material they'll be looking at, even their probable research questions, in a way them telling me they're studying "4000 to 2500 BC" wouldn't. But then I might have to ask them, "so when's your Neolithic? 3000-ish?" And if it is 3000-ish, should I really be linking what they study with the "Neolithic" in other places, where it might have started in 6000 or 9000? I don't think so, and where there are robust radiocarbon chronologies available lots of archaeologists prefer to use absolute date ranges in published work, even if privately, out of convention, we continue to talk and think in ages.