The first part of this thesis deals with the broad problem of forming the optimum estimate of an analog message which has been transmitted over a channel that can be modelled by a linear randomly-timevarying filter and additive white Gaussian noise. A two-step approach to the evaluation of the conditional mean is proposed. First, the optimum message estimate is evaluated under the assumption that the channel is known exactly. Secondly, the optimum estimate of this conditional estimate is evaluated. An estimate of the channel or some gain that depends on the channel is formed during this second step. It is shown that the mean square error of the optimum message estimate can be expressed in terms of two additive components. The first component is the mean square error of the optimum message estimate in the ideal situation where the channel can be measured exactly. The second component is the mean square error associated to the estimate of the channel. In the second part of this thesis, the particular problem of forming an estimate of the amplitude of a Gaussian pulse which has been. transmitted over a purely-fading channel is investigated in the context of nonlinear estimation theory. The performance of a physically realizable maximum aposteriori probability (MAP) estimate is evaluated. A transmitted reference system (TRS) is proposed in which part of the transmitted energy is devoted to a channel test signal. The unconstrained minimum mean square error estimate of the message is derived. The optimum receiver consists of a match filter followed by a gain which depends nonlinearly on the unconstrained minimum mean square error estimate of the channel. The performance of the TRS is . evaluated and the optimum energy division between the message signal and the channel test signal is investigated.