If you have had problems with drainage in the past then perhaps you have been using a natural slope for your existing lawn. When you make the switch to synthetic turf, this is no longer an issue.
This means that you have the option of removing the slope and getting a beautifully flat lawn if you want to. On the other hand, if you want to carry on with the same sloping lawn, then this is also an option.
You won’t have any problems with installing artificial grass on a slope, although you will want to be sure to fix it down well if the slope is steep. Fixing it with nails and avoiding using a sand infill are a few techniques you might need to use.
Generally speaking, a gradient of 15% or less is necessary for walking on, while a steeper slope means that you will probably use your lawn for purely decorative purposes. Of course, no matter what type of grass you use, you will also want to ensure that you aren’t just sending the excess rainwater somewhere else to gather and cause puddles to form.