The ability to perform efficient and stable micro-canonical simulations with molecular trajectories evolving on the Born-Oppenheimer potential energy surface is discussed. Until recently this was possible only by relying on expensive high-precision calculations. With the introduction of time-reversible Born-Oppenheimer molecular dynamics last year this problem was solved. I will review the original framework of time-reversible Born-Oppenheimer molecular dynamics and present some recent developments based on an extended Lagrangian formulation. The extended Born- Oppenheimer Lagrangian provides a more rigorous, versatile, and physically transparent framework compared to the previous formulation. The extended Born-Oppenheimer Lagrangian also allows for a direct comparison to the original extended Lagrangian approach for ab initio molecular dynamics simulations invented by Car and Parrinello over 20 years ago.