Acknowledgements; Introduction; 1. Simple groups and indecomposable subgroups of GLn(2); 2. Dickson group G2(3) and related simple groups; 3. Conway's simple group Co3; 4. Conway's simple group Co2; 5. Fischer's simple group Fi22; 6. Fischer's simple group Fi23; 7. Conway's simple group Co1; 8. Janko group J4; 9. Fischer's simple group Fi'24; 10. Tits group 2F4(2)'; 11. McLaughlin group McL; 12. Rudvalis group Ru; 13. Lyons group Ly; 14. Suzuki group Suz; 15. The O'Nan group ON; 16. Concluding remarks and open problems; Appendix: Table of contents of attached DVD; References; Index.