Animals play a profoundly important and intricate role in our lives today. Dogs have been human companions for thousands of years, but now they work to assist the disabled, and in combat and search and rescue situations. Farm animals are a critical part of sustainable agriculture today, and there is increasing consumer interest in 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 habitats. There are many reasons, including societal and economic ones, to explore how new computing technologies can be used to ensure the welfare of animals in these settings. The goal of this review is to systematically survey the existing literature in smart computing and sensing technologies for domestic, farm, and wild animal welfare. We use a broad notion of animal welfare to refer to an assessment of whether animals are healthy, free of pain and suffering, and positively stimulated in their environment. Smart computing and sensing is also used in broad terms, to refer to systems that are not isolated but interconnected with communication networks, and capable of remote data collection, processing, exchange, and analysis. The findings of this review are expected to motivate future research in computer science and engineering, as well as contribute to data, information, and communication management for animal welfare.