What does “good” mathematics look like? Certainly, it should be correct, but mathematicians are often far more demanding. For example, they want their work to be deep, explanatory, fitting or even beautiful. This simple observation from mathematical practice raises philosophical questions: What do these terms mean? Why is work that exhibits these properties valuable? Are there design principles we can follow to help ensure our mathematics has these qualities? Throughout this course we will explore these questions by seeing what mathematicians and philosophers have had to say and by examining both modern and historical case studies.