Curiosity is sometimes conceptualized as a psychological state that includes recognition of a knowledge gap, anticipation that it may be possible to close the gap, and an intrinsically motivated desire to do so. Conceptualized in this way, curiosity can be theoretically linked to mathematical problem solving and problem posing. In this talk, I attempt to explore this link in didactic situations that combine problem posing and problem solving as one multi-stage activity. I will argue for three possible roles for curiosity: curiosity as a manifestation of learning autonomy that can be captured as a basis for problem posing and problem solving, curiosity as an emotional state that can be triggered during problem posing and then mediate it, and curiosity as a human trait that can be developed by means of repeated experience of problem posing and problem solving. I will illustrate these roles of curiosity by means of the data collected in a M.Sc. course for in-service mathematics teachers that I led in 2022.