This work is an attempt to see how physics undergraduates view the basic ideas of general relativity when they are exposed to the topic in a standard introductory course. Since the subject is conceptually and technically difficult, we adopted a "case studies" approach, focusing in depth on about six students who had just finished a one semester course on special relativity. The methodology of investigation involved a combination of text comprehension questionnaire and detailed clinical interviews. The aim was not to investigate the technical proficiency of the students, but to probe in detail the nuances of their conceptions of several basic points of the subject. Analysis of their responses reveals a large number of "alternative conceptions" of students in the domain. The study should be useful to physics education researchers as well as to teachers of introductory general relativity at about the senior undergraduate level. [A 23-item list of references and notes is provided.]