Cats do sweat, but only in very limited areas. Most cats only sweat through the bottom of their paws. Sweat is not an effective way for cats to regulate their body temperature.
Panting is the most effective way for an overheated cat to cool itself down. Cats may sweat or pant in very hot weather or if they are stressed or sick. Cats also regulate their body temperature by resting on cool surfaces, stretching out to allow their body heat to dissipate and getting their bodies wet. Cats that are panting excessively or that cannot seem to cool themselves down should be seen by a veterinarian.