The paper deals with the application of the feed-forward and cascade-forward neural networks to mechanical state variable estimation of the drive system with elastic coupling. The learning procedure of neural estimators is described and the influence of the input vector size and neural network structure to the accuracy of state variable estimation is investigated. The quality of state estimation by neural estimators of different types is tested and compared. The simple optimisation procedure is proposed. Optimised neural estimators of the torsional torque and the load machine speed are tested in the open-loop and closed-loop control structure of the drive system with elastic joint, with additional feedbacks from the shaft torque and the difference between the motor and the load speeds. It is shown that torsional vibrations of the two-mass system are damped effectively using the closed-loop control structure with additional feedbacks obtained from the developed neural estimators. The simulation results are confirmed by laboratory experiments.