Training in research methods is a crucial component of the student experience in further and higher education. A common set of statistical and experimental design methods are taught across a broad range of non-mathematics disciplines, spanning STEM subjects, medicine, and the social sciences. Understanding these methods is central to students' ability to engage with their course, tutors, and the literature. It is also the key to enabling students to become not only practitioners of their chosen subject, but also statistically literate citizens, capable of understanding and evaluating everyday statistics. The first aim of this paper is to review the specific set of challenges faced by staff and students teaching and learning statistics within non-mathematics disciplines. Secondly, we review best practice and current trends in the design of motivating and effective statistics courses for non-specialists. Our findings suggest that many of the key challenges stem from negative attitudes towards statistics coupled with poor motivation to study the subject, factors which are exacerbated by statistics anxiety. Fortunately, because these challenges are so widespread, and have attracted the attention of innovative educators across broad disciplines, there is a wealth of good ideas and resources available to statistics teachers seeking ways to create effective learning experiences.