This study is based on our belief that mathematics should be challenging in any classroom and that mathematical challenge is among the central factors that determine the quality of mathematics lessons. Choosing challenging mathematical problem for the students is central in teachers' work while their conception of mathematical challenge can determine the quality of a mathematics lesson. At the same time, little is known about teachers' views on mathematical challenge. Thus, we explored conceptions of mathematical challenge in two groups of experienced mathematics teachers. The first group (N[subscript 1] = 9) was asked to define the notion of mathematical challenge and give examples of challenging mathematical tasks. Later the members of the group discussed these examples and definitions. A written response questionnaire was administered to a second group of teachers (N[subscript 2] = 41) based on answers given by teachers in the first group. We found that the teachers have a broad conception of mathematical challenge and appreciate the relativity of mathematical challenge but are not always convinced that it is possible to incorporate challenging mathematics in everyday teaching in the classroom.