Introduction; Part I. The Basic Theory: 1. Admissible set theory; 2. Some admissible sets; 3. Countable fragments of L¿¿; 4. Elementary results on HYPM; Part II. The Absolute Theory: 5. The recursion theory of ¿1, predicates on admissible sets; 6. Inductive definitions; Part III. Towards a General Theory: 7. More about L¿¿; 8. Strict ¿11 predicates and Koenig principles; Appendix. Nonstandard compactness arguments and the admissible cover; References; Index of notation; Subject index.