Unirary infections is the big one for cats. It seems it their sensitive point. My vet told me as well that some cats are more prone to it than others. I have an "orange cat" and she told me to try to have him drink a lot of liquid to protect his kidneys. So each evening, I give him some "cat milk" diluted with water and he loves it.
Overweight cats develop easily diabetes as well. May be it's like human as well. I remember that once my neighbours asked me to take care of the cat of their daughter who while they were all on holidays and I had to give an injection twice a day to the cat.... and he knew it was good for him.
When cat become older (15 and older), they seem to develop mouth cancer. My three cats who past away all developped that; one was 15 years old, another 16 years old and the last one 18 years old.