This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.