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 blockage or repair in a drain that is shared with one of your neighbours is also up to the water authority to remedy, even if the drain is still on your property.
Why does it matter who is responsible?
If it is the responsibility of your local water authority, then they will clear blocked drains free of charge.
Still confused? Call 0161 885 3775 for free advice.