Could you specify which culture/region you're interested in? Answers will vary widely. To get you started, here's a recent similar post

• As wives traditionally take their husbands' surnames, does that mean there are fewer surnames than in the past?

Here's the FAQ section on the development/adoption of surnames, which I include here to illustrate the incredibly wide range of dates that surnames have been adopted (or even not yet adopted) in various cultures.

• Surnames