It is probably easier to say what drains are not your responsibility, rather than what drains are down to you.
Any problems with drains off the boundary of your property, are up to your local water authority to clear.
A collapsed or broken drain that is shared with one of your neighbours is also up to the water authority to repair, even if the drain is actually on your property.
Why does it matter who is responsible?
If it is the responsibility of your local water authority, then they will make good any repairs free of charge.