In this symposium volume twenty-nine key figures in the classification programme analyse the likely directions research will take in the future.With the classification of finite groups an era of research in that subject ended. In this symposium volume twenty-nine key figures in the classification programme analyse the likely directions research will take in the future.With the classification of finite groups an era of research in that subject ended. In this symposium volume twenty-nine key figures in the classification programme analyse the likely directions research will take in the future.With the classification of finite groups an era of research in that subject ended. In this symposium volume twenty-nine key figures in the classification programme analyse the likely directions research will take in the future.1. Computation; 2. Using the Classification; 3. Properties of K-Groups; 4. Groups and Algebras; 5. Local Analysis; 6. Amalgams and Finite Geometries; 7. Representation Theory; 8. Rigidity and Galois Groups; 9. Other Connections with Number Theory; 10. Other Directions.