People around the world revered animals, as manifestations of their gods, as votive objects, and as beloved pets. In ancient Egypt, embalmers treated some animals, like the Apis bull, with as much care as many humans. Here's one at the Smithsonian Institution:
Cat mummies abound, as do ibises (sacred offerings to the god Thoth).
But in Peru, some animals were buried as pets, like this naturally dessicated puma with claws that were unworn. It was buried with jewelry and a special mantle:
Other animal mummies are just accidents of nature in a dry climate, such as this shriveled-up iguana I recently saw in the Galapagos:
My new favorite animal (not a mummy, just a burial) is a bobcat kitten found in Illinois. Buried with a necklace, it was clearly a valued pet until it died. Here's a reconstruction by Ken Farnsworth: