This dissertation demonstrates the innovative use of neural networks for the estimation and control of vibrations in large, flexible space structures. As reported in the literature, the control of vibrations in this distributed parameter system is challenging due to the dual problem of observer and controller spillover when a discretized, finite dimensional model and traditional control system methods are used. The neural network based method for both modeling and control as developed in this research overcomes the observer and controller spillover because the method is adaptive. An estimator for prediction of the next values of acceleration at locations where accelerometers are located along a structure is developed using the ADALINE neural network based model and a tapped delay line. The estimator's output, which represents acceleration at these locations, is compared to the output of this large space structure which is based on an analytic model. Errors are consistently less than 15% of the testbed reference model's output at higher frequencies, and even less at lower frequencies. A control system is developed using neural networks operating on a decentralized PID feedback scheme. The neural network has an on/off, bidirectional output to mimic the operation of the control thrusters on the testbed facility. The network learns by changing the magnitude of the output function and the on/off deadband, rather than the coefficients of the activation function. The control system consistently damps vibrations by a factor of ten within two seconds. This performance compares favorably with other currently utilized methods.