Animals play a profoundly important and intricate role in our lives today. Dogs have been human companions for thousands of years, but they now work closely with us to assist the disabled, and in combat and search and rescue situations. Farm animals are a critical part of the global food supply chain, and there is increasing consumer interest in organically fed and humanely raised livestock, and how it impacts our health and environmental footprint. Wild animals are threatened with extinction by human induced factors, and shrinking and compromised habitat. This review sets the goal to systematically survey the existing literature in smart computing and sensing technologies for domestic, farm and wild animal welfare. We use the notion of animal welfare in broad terms, to review the technologies for assessing whether animals are healthy, free of pain and suffering, and also positively stimulated in their environment. Also the notion of smart computing and sensing is used in broad terms, to refer to computing and sensing systems that are not isolated but interconnected with communication networks, and capable of remote data collection, processing, exchange and analysis. We review smart technologies for domestic animals, indoor and outdoor animal farming, as well as animals in the wild and zoos. The findings of this review are expected to motivate future research and contribute to data, information and communication management as well as policy for animal welfare.