This course will introduce students to proof theory, the study of formals proof. We will compare different proof systems, such axiom systems, natural deduction, and sequent systems, and we will learn how to prove things about these systems. Students will learn the major foundational results in the area, Normalization and Cut Elimination. Once we have learned the basic results, we will look at proof systems for non-classical and modal logics, to see how one can adapt the basic frameworks for these new logics. Throughout the course, we will see ways in which proof systems themselves are philosophically significant and what they can tell us about the meaning of logical connectives.