Attention bias describes the differential allocation of attention towards one stimulus compared to others. In humans, this bias can be mediated by the observer's affective state and is implicated in the onset and maintenance of affective disorders such as anxiety. Affect-driven attention biases (ADABs) have also been identified in a few other species. Here, we review the literature on ADABs in animals and discuss their utility as welfare indicators. Despite a limited research effort, several studies have found that negative affective states modulate attention to negative (i.e., threatening) cues. ADABs influenced by positive-valence states have also been documented in animals. We discuss methods for measuring ADAB and conclude that looking time, dot-probe, and emotional spatial cueing paradigms are particularly promising. Research is needed to test them with a wider range of species, investigate attentional scope as an indicator of affect, and explore the possible causative role of attention biases in determining animal wellbeing. Finally, we argue that ADABs might not be best-utilized as indicators of general valence, but instead to reveal specific emotions, motivations, aversions, and preferences. Paying attention to the human literature could facilitate these advances.