A gcneral type of forward-backward doubly stochastic differential equations (FBDSDEs) is studied. It extends many important equations that have been well stud-led, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a method of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.