Online estimation of the parameters of an exponentially damped sinusoidal (EDS) signal is a classical problem in systems theory that has some practical applications. Examples include audio signal modeling and system identification. Audio signals contain relatively fast variations in amplitude and cannot be efficiently modeled as constant-amplitude sinusoids. In system identification problems, the characterization of the impulse response of a linear system by a sum of complex exponentials and then identifying or approximating the complex amplitudes and the natural frequencies with high degree of accuracy is of special importance. Parameter estimation of an EDS signal consists of amplitude, damping factor, frequency and phase of the signal. Nonlinear dependence of an EDS signal on its frequency and damping factor makes the estimation of these two parameters a challenging theoretical problem. In this thesis, an adaptive algorithm is proposed for the estimation of the frequency and damping factor of an EDS signal. The algorithm is based on an adaptive identifier which is a theoretical tool for the estimation of the parameters of an unknown linear system. The governing equations of the proposed algorithm are composed of a second order linear filter and two update laws that provide indirect estimation of the frequency and damping factor. Since the EDS signal is a finite duration signal, the parameters which are estimated by the standard gradient method, are not guaranteed to converge to their nominal values. To guarantee the convergence, the parameters update laws derived by the gradient algorithm are properly normalized. Local asymptotic stability of the proposed algorithm is carried out based on the averaging theory. Various aspects of the proposed algorithm are investigated through extensive computer simulations. Specifically, a number of simulations are conducted to show that the proposed method is appropriate for estimation of the parameters that are slowly time varying. Moreover, it is shown that the algorithm is capable of estimating the frequency and damping factor of an exponentially undamped sinusoidal signal and estimating the frequency of a sinusoidal signal. The thesis further contributes in presenting an adaptive algorithm for the rejection of the EDS type disturbances affecting a linear and time invariant system. For this, the proposed algorithm for the estimation of frequency and damping factor of an ESD signal is used. The proposed compensator which is based on the concept of the adaptive feedforward cancelation (AFC) scheme reproduces the disturbance signal in the loop. The advantage of the proposed compensator is that the frequency and damping factor of the disturbance signal is estimated independently of the cancellation scheme for an EDS signal with known frequency and damping factor. The proposed algorithm is simple in structure and can be easily designed, as long as the plant is known and stable. Based on simulation examples, the performance of the compensator is evaluated. Compared with the internal model principle algorithm, the parameters tuning in the proposed method is simpler. Fast convergence rate of the parameters and rejection of the disturbance signal in short time are the other salient features of the system. Simulation results confirm the analytical derivations and the desirable performance of the proposed compensator. Key Words: Exponentially damped sinusoidal signal, Frequency, Damping factor, Adaptive identifier, Averaging theory, Adaptive feedforward cancellation