We provide an overview of how the notion of randomness is treated in mathematics and in mathematics education research. We then report on two studies that investigated students' perceptions of random situations. In the first study, we analyze responses of prospective secondary school teachers who were asked to provide examples of random situations. In the second study, we focus in depth on participants' perceptions of randomness in a clinical interview setting. Particular attention is given to the participants' ways of communicating the idea of randomness, as featured in the gestures that accompanied their discourse. We conclude that particular consideration of the notion of randomness—as intended in statistics and probability versus everyday uses of the term—deserves attention of instructors and instructional materials.