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.