Questions about the nature and causal relationships in the universe include the "arrow of time", the initial conditions for the birth of the universe, the absorption of information by black holes, etc. The discussion between Hawking and Penrose is largely a continuation of the Solvay debate between Bohr and Einstein on the basics of quantum mechanics. For a wide audience of readers interested in the problems of modern physics and cosmology Preface Gratitude Chapter 1. Stephen Hawking . Classical Theory Chapter 2. Roger Penrose . Structure of space-time singularities Chapter 3. Stephen Hawking . Quantum Black Holes Chapter 4. Roger Penrose . Quantum Theory and Space-Time Chapter 5. Stephen Hawking . Quantum Cosmology Chapter 6. Roger Penrose . A Historical View of Space-Time Chapter 7. Stephen Hawking and Roger Penrose . Discussion of literature