This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory.The following topics are covered:* Forcing and constructability* The Solovay-Shelah Theorem i.e. the equiconsistency of 'every set of reals is Lebesgue measurable' with one inaccessible cardinal* Fine structure theory and a modern approach to sharps* Jensen's Covering Lemma* The equivalence of analytic determinacy with sharps* The theory of extenders and iteration trees* A proof of projective determinacy from Woodin cardinals.Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers.