Current reforms in undergraduate mathematics education devoted to proof and counterexample highlight its importance in teaching and learning mathematics. While undergraduate students who have taken a number of upper-level mathematics courses are expected to have mastered the skills required to verify statements as well as to read and write proofs and counterexamples, the literature shows that those students still have considerable difficulty with these tasks. To date, however, little research has focused on undergraduates' strategies for validating arguments, evaluating statements, and constructing proofs and counterexamples in different contexts. Thus, the central question of this dissertation is to understand the strategies undergraduate students have for doing such tasks in the domains of algebra, analysis, geometry, and number theory. This study focused primarily on processes through which undergraduate mathematics majors: (1) validated given arguments as correct proofs or counterexamples; (2) evaluated the truth and falsity of statements; and (3) produced a proof or a counterexample, depending on whether they believed the statement to be true or false. In order to explore undergraduate students' conceptions of proof and counterexample, this study also examined their understandings of what constituted proof and counterexample. To answer the aforementioned questions, the data from 16 undergraduate mathematics majors, including eight specializing in secondary mathematics education, were gathered through a series of interviews in which they were asked to evaluate the validity of various proofs and counterexamples or produce proofs or counterexamples to various mathematical claims. Findings from this study suggest that most undergraduate students are generally confident in their understanding of the mathematics, yet they often struggle both in identifying valid proofs and counterexamples and in producing valid proofs and counterexamples. This study also suggests that some undergraduate students possess insufficient mathematical knowledge of reading and writing proofs and counterexamples as well as have inadequate understandings of what constitutes proof and counterexample. Implications for undergraduate mathematics education resulting from this study include a need to develop undergraduates' skills in reading and writing proofs and counterexamples and to enhance their conceptions of proof and counterexample. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]