1. Summary -- 2. Heights -- 3. Normalised heights -- 4. The Mordell-Weil theorem -- 5. Mordellโ{128}{153}s conjecture -- 6. Local calculation of normalised heights -- 7. Siegelโ{128}{153}s method -- 8. Bakerโ{128}{153}s method -- 9. Hilbertโ{128}{153}s irreducibility theorem -- 10. Construction of Galois extensions -- 11. Construction of elliptic curves of large rank -- 12. The large sieve -- 13. Applications of the large sieve to thin sets -- Appendix: The class number 1 problem and integral points on modular curves -- A.1. Historical remarks -- A.4. Elliptic curves with complex multiplication -- A.5. Modular curves associated to normalisers of Cartan subgroups and their CM integral points -- A.6. Examples -- A.7. The Gelโ{128}{153}fond-Linnik-Baker method