In 1931, Princeton mathematician Kurt G"odel startled the scientific world with his Theorem of Undecidability, which showed that some statements in mathematics are inherently `undecidable'. This volume of the Oxford Logic Guides is a sequel to Smullyan's G"odel's Incompleteness Theorems (Oxford Logic Guides No. 19, 1992), but it can be read independently by anyone familiar with at least one proof of G"odel's incompleteness theorem for Peano Arithmetic. This volume was written for mathematicians, philosphers, and computer scientists who are curious about G"odel's incompleteness theorems. The only prerequisite would be a first course in mathematical logic.