It depended on the culture. In Iceland today as of 20 years ago, inherited surnames ("last names") are not done except for some grandfathering (no pun intended). In England, my source (Reaney and Wilson, citing Fransson) suggests that hereditary surnames were in the nobility in the 12th century, in the general populace fairly frequently by the end of the 13th century, and general at the end of the 14th century. In Wales, they suggest inherited surnames started in the early 1500s from the gentry, but literal surnames (where Roberts meant that your father really was named Roberts) was not uncommon in Merionethshire even in the 19th century. For Ireland: which culture and language, Gaelic or Anglo-Irish? For Gaelic, clan surnames are kind of hard to classify: you were Uí Néill and your father was a Uí Néill, but you're both (presumably) descendents of a Niall, so it's both inherited and literal. In general in Ireland, the conversion to clear inherited surnames was gradual. Scotland has the same situation, lowlands (Scots, English-influenced) versus highlands (Gaelic), and I don't have firm dates for Gaelic (whether in Scotland or Ireland), but for Scots, almost everyone by 1500.

If you have some particular European culture, language, and/or place in mind, I may be able to rustle up an expert who can address it.