1. Introduction -- References -- 2. Fundamentals -- 2.1 The rules of the game -- 2.2 The whole and its parts -- 2.3 Classical limit and decoherence -- 2.4 Proofs of the frequency interpretation -- References -- 3. How ? Collapses -- 3.1 The problem -- 3.2 Collapse and unitary evolution -- References -- 4. Bell's Inequalities -- 4.1 The Einstein, Podolski and Rosen (EPR) argument -- 4.2 D'Espagnat's version of Bell's inequalities -- 4.3 Stapp's formulation of Bell's inequalities -- 4.4 Bell's proof -- References -- 5. Copenhagen Interpretation -- 5.1 Absence of quantum objects -- 5.2 Objectively existing potentialities world -- References -- 6. Many Worlds Interpretation (MWI) -- 6.1 The role of the observer -- 6.2 Nonlocality -- 6.3 Criticisms -- References -- 7. Quantum Logic (QL) -- 7.1 Proposal -- 7.2 Hasse diagrams -- 7.3 The birth of time -- References -- 8. Macroscopic Realizations of QL -- 8.1 Lattices and graphs -- 8.2 Breakdown of Bell's inequalities -- 8.3 The Problem of Nonlocality -- References -- 9. Hidden Variables Theories (HVT) -- 9.1 What is a hidden variables theory? -- 9.2 Von Neumann's theorem -- 9.3 Spin 1/2 hidden variable's model -- 9.4 Gleason's theorem. Kochen-Specker paradox -- References -- 10. De broglie-Bohm Non. Relativistic HVT -- 10.1 Non relativistic de Broglie-Bohm theory -- 10.2 The many particle system -- 10.3 Remarks on the non relativistic BBT -- References -- 11. De Broglie-Bohm Relativistic HVT -- 11.1 HVT interpretation of the boson equations -- 11.2 The BBT interpretation of the Dirac equation -- 11.3 Relativistic BBT many-fermion theory -- 11.4 Necessity of superluminal processes -- References -- 12. Statistical Interpretation -- 12.1 Introduction -- 12.2 When is the SIQT possible? -- 12.3 SIQT × orthodox quantum theory -- References -- 13. Non-Unitary Evolution? -- 13.1 Prigogine and Ghirardi et al. theories -- 13.2 Criticisms -- References -- 14. Histories Approach -- 14.1 What is a history? -- 14.2 Consistent histories -- 14.3 Criticisms -- References -- 15. Quantum Miracles and Cryptography -- 15.1 Greenberger et al. example -- 15.2 Hardy's example -- 15.3 Negative experiments. Elitzur-Vaidman proposal -- 15.4 Delayed choice experiments -- 15.5 Quantum erasers -- 15.6 Quantum cryptography -- 15.7 Teleportation -- References -- 16. Where are We Now? -- References -- Appendices -- A- Set Theory and Lattices -- A.1 Main definitions -- A.2 Posets -- A.3 Lattices -- A.4 The Boolean algebra of sets -- A.5 Borel sets, measures and integration -- References -- B- Hilbert Spaces -- B.1 Definition of Hilbert Space -- B.2 Linear manifolds and subspaces -- B.2.1 The lattice of subspaces -- B.3 Dual and conjugate dual spaces -- B.4 Tensor products -- B.5 Linear operators -- B.6 Projection operators and lattice structure -- References.