Despite the unarguable importance of Stack Overflow for the daily work of many software developers and the existing knowledge about the impact of code duplication on software projects, the prevalence and implications of code clones on Stack Overflow have not yet received the attention they deserve. In this paper, we motivate why studies of this aspect are needed and how existing studies on code reuse from Stack Overflow differ from this new research direction. We present similarities and differences between code clones in general and code clones on Stack Overflow and point to open questions that need to be addressed to be able to make data-informed decisions about how to handle clones on this important platform. We present results from a first preliminary investigation indicating that clones on Stack Overflow are common and diverse and conclude with possible directions for future work.