I am an active mathematical physicist who has also engaged long-term with mathematics education; particularly, with research on mathematical learning and problem solving. This led, perhaps inevitably, to a focus not only on cognition, but also on the psychology of what is called “the affective domain” – i.e., emotional feelings, attitudes, beliefs, and values – in relation to mathematics. In this talk, I shall discuss some important affective constructs which relate directly to mathematical teaching and learning, with a particular focus on the nature of student engagement. Among the ideas considered are the importance of emotional feelings during mathematical problem solving, the idea that information important to learning is encoded affectively (interactions with cognition), the role of beliefs about mathematics, students’ (longer-term) motivational orientations, and various “in the moment” motivating desires that can foster (or inhibit) students’ mathematical engagement, as well as difficulties that arise in efforts to study mathematical affect. Some broader implications are suggested for how we prepare mathematics teachers, how we connect with our own students, and how we represent mathematics to the wider community.