I can recommend reading some earlier threads while you are waiting. The Ancient Greeks did not have surnames as we know them today, and when they had to they used forms like "Herodotus the Halicarnassian" or "Xenophon son of Gryllus". You can read more about that in answers like this one by u/XenophonTheAthenian, this one by u/Iphikrates and this one again by XenophonTheAthenian.

This pattern seems reasonably common in the Ancient Mediterranean: for instance in this answer by u/Georgy_K_Zhukov and this one by u/Trevor_Culley mentions patronymics in ancient Jewish names, and according to these threads (1, 2, 3) by u/Scipio_Asina it was also prevalent in Carthage.

On the other hand the Romans (and some other Italian peoples) did have a system with several names (stereotypically three) of which two could be considered family names. This has been discussed a lot on here. For instance in this old thread u/Celebreth does a basic rundown of the Roman system with u/heyheymse chiming in with important contributions about women's names, and Celebreth has also written more about how names would be used here. Some complications, and how this system changed over time, are elaborated by u/Astrogator here and here, and by our XenophonTheAthenian in this thread. Further answers by our users are of course appreciated!