Epileptic seizures are manifestations of epilepsy, a serious brain dynamical disorder second only to strokes. Of the world's approximately 50 million people with epilepsy, fully 1/3 have seizures that are not controlled by anti-convulsant medication. The field of seizure prediction, in which engineering technologies are used to decode brain signals and search for precursors of impending epileptic seizures, holds great promise to elucidate the dynamical mechanisms underlying the disorder, as well as to enable implantable devices to intervene in time to treat epilepsy. There is currently an explosion of interest in this field in academic centers and medical industry with clinical trials underway to test potential prediction and intervention methodology and devices for Food and Drug Administration (FDA) approval. This invited paper presents an overview of the application of signal processing methodologies based upon the theory of nonlinear dynamics to the problem of seizure prediction. Broader application of these developments to a variety of systems requiring monitoring, forecasting and control is a natural outgrowth of this field.