Self-driving cars (aka Driverless Cars) are being researched widely by different companies. The Nevada State was the first one to allow such cars to be tested in the roads. Google is behind this achievement.
Similarly to Uber and some other apps that impact the way we live, each country will have it's fair time of court to debate whether driverless cars are allowed or not. Without a specific jurisdiction in mind, your question is too-broad, but I believe that there will be countries with early authorization of such cars, whereas some countries will still take up a few decades to start introducing this new technology.
Up to this day, it is beyond the scope of my knowledge any driverless car in which the company states that a human driver is not required in the driver seat. Considering that it's a brand new technology and the outcomes are still not quite definitive, it's safer to require a human ready to act up and it shields the company from liability. We know the cars are being tested and we know drivers are required to be focused, so if an accident happens the driver is liable.
Here is a map of the USA jurisdiction about autonomous cars:
Currently, there are news about USA, UK, France and Switzerland allowing such cars to be tested in public roads.