far as the "oldest" last name. That is very difficult to answer. The Chinese have been using family names (although they say them before the personal name, instead of after) since around 2,800 BCE. Supposedly this was decreed by Emperor Fushi, last names having previously been something used only by the aristocratic or very rich. So, in all likelihood, the first "last name" was Chinese.
Surnames in the western world developed independantly of the Chinese practice. During the Roman Empire many families and individuals took surnames ("cognomen" or "nomen"). This could be very confusing, because although they were often passed down, they were not always. Sometimes people were known by an adjective or named for a deed they had performed, rather than by a family name.
The Roman practice of using family names died out as the Empire collapsed and, during much of the early Medieval period, most people (especially peasants) went by only one name. By about 1100, however, through practicality and the influence of the Venetian court, surnames had once again spread throughout much of Europe.
Between that time period and about 1450 names were still evolving and not always fixed. By the 15th century, however, most people of all ranks had some sort of fixed, hereditary surname.
There are some exceptions, of course. Many Scandanavian countries continued patronymic naming well into the the 19th century. "Patronymic" means named after the father, which doesn’t translate to a fixed family surname. Lars, son of Eric would be Lars Ericson, his son might be Peter Larsson, his son Eric Peterson, and so on. (In some regions females used the suffix -datter, so Lars’s daughter Hilde would be Hilde Larsdatter. Those obviously don’t survive as long).
The practice of family last names spread slowly into the Baltic countries and firther eastward. Some cultures resisted. In Turkey, for example, last names weren’t common until they were mandated in 1933 by a ruler attempting to "modernize" Turkey’s image! yes its Mustafa Kemal
Sources: Danesi, Marcel (2007). The Quest for Meaning
What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700," by Benet Salway; The Journal of Roman Studies, (1994)
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