The former owner of our house installed a catch basin in this area with a drainage pipe that runs underground to a location at the back part of the yard. This was a good idea and usually takes care of the problem, but in heavy rains, the catch basin overflows. To take care of this problem, we installed a dry streambed to take the excess water down the slope in a more controlled way.
As you can see from this picture, the streambed makes a turn toward the fence and disappears behind shrubs.
We did a similar thing in the side yard to direct run-off from the neighbor's driveway toward the street. Water collects in a low spot between the neighbor's house and ours (assisted by a berm that Jack created) and then enters a dry streambed in the area shown below.
We just finished the last section of this streambed this summer and have had almost no rain since. Consequently, we don't know for sure whether the discharge from the stream will stay to the side lawn as we hope. We will be closely watching this area throughout the winter and if we find low places that allow the water to flow toward the lawn rather than down the hill, we will mark those places. Then, in the spring, we will lift the sod there and create a slight berm to encourage the water to flow toward the street.
I have found doing this work ourselves to be very satisfying (Jack might disagree . . . ), and I really don't mind that it takes us a longer time to get a finished product. Being amateurs, we seldom get it perfectly right on the first attempt and have to do subsequent tweaking. By my calculations, we will finish all our garden projects, about the time we're ready for the retirement home. But, for gardeners, that might be perfect timing!!