As mathematics educators we want our students to develop a natural curiosity that will lead them on the path toward solving problems in a changing world, in fields that perhaps do not even exist today. Here we present student projects, adaptable for several mid- and upper-level mathematics courses, that require students to formulate their own questions and to begin to develop the basic research skills needed to answer these questions. These projects, where each student is given an individualized object to study, allow students to take ownership over their own learning while introducing them to the joy and challenge of discovery and research. Each student is directed to use the concepts and techniques presented in class as a set of tools to guide the investigation of their object. We discuss our experiences--both positive and negative--with these inquiry-based projects.