In this study we investigate the conceptions of mathematics bridging courses held by students enrolled in these courses at a major Australian university. We report on the participants' responses to email-interview questions about the mathematics bridging courses to describe a two-dimensional outcome space of variations in awareness about the bridging courses. On one dimension the conceptions relate to cognitive functions: the course bridges students' difficulties with mathematical concepts, helps develop strategies for learning mathematics and extends skills in thinking and reasoning. Categories on the other dimension reflect ideas on how the bridging course advances personal goals and enhances self-development. The findings show that students are aware of the value of the bridging courses not only to ameliorate prior difficulties with mathematics and improve their approaches to learning mathematics but, less transparently, as an important opportunity to facilitate their transition into higher education, meet fellow students and help realise their potential. (Contains 1 figure.)