<p>We establish the global-in-time well-posedness for a broad class of mean field games including those with the small mean field sensitivity and the linear-quadratic setting as special cases. Instead of using the master equation approach, we adopt the maximum principle to investigate the unique existence of the equilibrium strategy by solving the corresponding forward-backward stochastic differential equations (FBSDEs), whose global existence is shown by controlling the sensitivity of the backward solutions with respect to the initial data via new <i>a priori</i> estimates for the corresponding Jacobian flows. Besides, we provide the state-of-the-art study with general cost functions having both quadratic growth and non-convexity in the state variable. We also impose the structural conditions on the cost functions but not on the Hamiltonian. The advantages of this framework are threefold: (i) the structural conditions can be easily verified; (ii) reduced regularity of cost functions suffices for the unique existence of equilibrium solutions compared to solving the master equations; and (iii) when the mean field effect is not small, the cost functions are not convex in the state variable, or there is lack of monotonicity of cost functions, an accurate lifespan for the local existence of the FBSDEs is still given, which is not small in general. Finally, we provide a counterexample to illustrate the ill-posedness of the mean field games when the small mean field effect and the contemporary monotonicity conditions are violated, this demonstrates numerically that our assumptions should be sharp.</p>