This chapter is concerned with existence and uniqueness of classical solutions to the master equation. The importance of classical solutions will be demonstrated in the next chapter where they play a crucial role in proving the convergence of games with finitely many players to mean field games. We propose constructions based on the differentiability properties of the flow generated by the solutions of the forward-backward system of the McKean-Vlasov type representing the equilibrium of the mean field game on an L2-space. Existence of a classical solution is first established for small time. It is then extended to arbitrary finite time horizons under the additional Lasry-Lions monotonicity condition.