What are the emotions? I will present a mathematically-based framework that distinguishes the dimensionality, distribution, and conceptualization of emotion-related responses. Guided by this framework, large-scale investigations of emotional experience and expression reveal that emotion-related responses are high-dimensional, involve gradients between categories traditionally thought of as discrete (e.g., ‘fear’, ‘disgust’), and cannot be reduced to widely used domain-general scales (valence, arousal, etc.). These findings have guided advances in machine learning and affective neuroscience. I will describe how we have used deep neural networks to investigate the meanings of distinct dimensions of facial expression by analyzing millions of hours of naturalistic video from around the world, and how brain imaging studies are beginning to uncover convergent evidence that nuanced emotional experiences are represented in specific regions along the cortical surface. Computational approaches have paved the way for more refined and nuanced answers to central questions about the meanings, algorithms, and mechanisms of human emotion-related responses.