The concept of a tangent is important in understanding many topics in mathematics and science. Earlier studies on students' understanding of the concept of a tangent have reported that they have various misunderstandings and experience difficulties in transferring their knowledge about the tangent line from Euclidean geometry into calculus. In this study, we examine students' understanding of the tangent concept from the viewpoint of generalization. The findings of the study reveal differences in students' generalizations regarding the tangent concept. We discuss the causes of these differences and provide a theory to account for them. In light of the results, some suggestions are made for improving the teaching of the concept of a tangent.