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The Roman's picked up the concept of an alphabet either, directly or indirectly, from the Greeks during the eigth century BC. Over time, they would evolve it into what we now think of as capital or uppercase letters. The Roman alphabet reached its final form sometime during the Republican era, although serifs would be added a bit later.
By the reign of the Emperor Trajan in the second century AD, the alphabet had certainly long been "finished". Nethertheless, letters would later be added to it during the Middle Ages (W and Z, for instance).
Roman handwriting differed significantly from inscriptional lettering. Over time, handwriting styles evolved into what we think of as various uncial styles, which later influenced the Carolingian Minuscule around the year 800. These letterforms would be paired with Roman inscriptional forms—"lowercase" the the older "uppercase"—during the Renaissance.
Roman, and later Christian, conquests spread the Roman alphabet far outward from Italy. Today, it is one of the most-used writing systems in the world.