We focus on the PID controller design for time-delay systems. All the controller gains ( kP , kI , and kD ) and the delay τ are treated as free parameters. Such a problem (involving totally four free parameters) is of theoretical as well as practical importance, but, it has not been fully explored. First, we will develop an algebraic algorithm to solve the complete stability problem w.r.t. τ . Consequently, for any given PID controller vector (kP,kI,kD) , the distribution of NU(τ) ( NU(τ) denotes the number of characteristic roots in the right-half plane) can be accurately obtained and the exhaustive stability range of τ may be automatically calculated. Next, a global understanding of the distribution of NU(τ) over the whole (kP,kI,kD) -space may be achieved and all structural changes regarding the NU(τ) distribution can be analytically determined. To achieve such a goal, a complete positive real root classification (for the auxiliary characteristic equation) will be proposed. Finally, we will give the procedure for determining the stability set in the (kP,kI,kD,τ) -space.