In this paper, several research topics are presented that provide productive avenues of exploration for students in an undergraduate cryptography course. The topics are intended for a variety of skill levels and interests. In some cases, a student might be inspired to try to solve the proposed problem or write computer code. Other topics might motivate them to learn new mathematics or algorithms, or research a historical event or person. In all cases, a short description of the topic is given along with several references that can be used as a starting point for student research. The author also describes some challenges and recommendations when teaching a course where a project is a main component.