Curiosity can take us inside a mathematical topic with which we are familiar or outside into mathematics new to us and even, if we hit a topic at the right time in the right way, new to the mathematical world. This paper includes over a dozen examples of curiosity, told as personal stories, organized by four main aspects of mathematical understanding; skills and algorithms; reasoning and proofs; applications and modeling; and language and representations. Some of the stories exemplify how curiosity led to my mathematical growth. Other stories influenced the curricular materials I was developing and writing for secondary school mathematics students. Content areas include arithmetic, algebra, geometry, functions, and discrete mathematics and their applications.