Although manure from vegetarian pets (cows, sheep, horses, gerbils, guinea pigs, rabbits, etc.) is considered safe for composting, manure from meat-eating pets such as cats and dogs can be problematic due to the risk of spreading zoonotic diseases (illnesses that can be transmitted from animals to people).
Some individuals and organizations do promote composting cat or dog waste, and there are even pet waste composters commercially available. However, if composting pet waste, set up a separate system for it (away from all food production gardens and water sources such as wells), and don’t apply the finished product near produce gardens or drinking water sources.
For answers to more frequently asked composting questions, see the Composting FAQ page. For a full list of gardening articles and links to garden photos, visit the main Gardening page.