Patronymics appear in many languages, but "true" patronymics are not very common. By "true" patronymics I mean names that is constructed from the father's name so that surnames change with every generation. The Gaelic "Macs" are not "true" patronymics in this sense, since Angus MacDougal's son Kenneth would be called Kenneth MacDougal, not Kenneth MacAngus. The name MacDougal, could be translated as "son of Dougal" but the surname actually means "of the clan of the son of Dougal"
In the Nordic countries surnames used to be constructed from the first name of one of the parents (usually the father), adding "son" or "daughter" as appropriate. If a Swedish man called Karl had a son named Leif and a daughter named Stina, the children would be known as Leif Karlsson and Stina Karlsdotter. If young Leif later on distinguished himself as a painter ("Målare" in Swedish) he might be known as "Leif Karlsson Målare", but the last part of his surname would not be inherited by his children.
Even if it's still perfectly legal, this naming practice is no longer the norm in Sweden, Norway or Denmark, but is still used on Iceland. Until recently Icelandic law stated that all citizens had to have a name conforming to the patronymic pattern, so immigrants had to come up with new Icelandic-sounding names.
In Russia another variant of patronymics is used where you both have a family name and a patronymic. E.g. Mikhail Baruskov has a son named Pjotr, who will be known as Pjotr Mikhailovich Baruskov. The variants and complexities of Russian surnames are mind-boggling and this was just a simple example. See "Russian patronymics" for more.
Thanks to gn0sis for pointing out that the most common patronymic suffix in Russia is "-ich" or "-ovich"