hmm over hanging your property ?
im sorry but to be able to give a better answer maybe you could elaborate on this more .. for example is it on your property does it connect with your guttering or does it help drain away your excess water to ?
if the guttering is attached with yours then the proportion on your property would be your responsibilty.
if its a joint pipework system the cost of repairs could be down to both parties " this type of system usualy has only one drain away pipe to the sewer between a number of properties "
if however its just their guttering thats " come adrift and therefore overhanging your property , then it is your neighbours responsibilty .
Either way something will need to be done as lack of drainage can cause a multitude of problems for example damp walls that later suffer mould , water can also cause structural problems given enough time with lack of drainage.