This book contains those topics necessary for postgraduates, research scholars and research programs of Indian and Foreign Universities chapter rst is introductory in nature and describe different types of recurrent neural network and briey explain sequence mapping systems, associative memories, architecture, Control theory assumption, Elman network and Jordan networks and Some limitation of gradient method of RNNS. Second chapter is devoted to Real Time Recurrent Neural Network and Learning, Fully Recurrent Neural Network, denition of Layers and Two Type of Fully Recurrent Neural Network. Chapter third explain Partially Recurrent Network State – space model and matrix notation. In chapter four we study Simple recurrent neural network , Layer denition, Architectures based on the input-output system model, Modular recurrent neural network architectures Recurrent multilayer perception, Block feedback networks, General modular network framework. Chapter ve focuses attention on Recurrent Neuro Dynamical System, Phase space Major forms, Equilibrium state ,Stability , Lyapunov's Theorem, State equation, Linear dynamical system, Non linear Dynamical system. In chapter six we summarised Long Short Term Memory Network ,Traditional LSTM, Learning, Limit of Traditional LSTM, the problem of Long-Term Dependencies. Some of our research work published in various journals has been discussed in the form of Chapter seven, and chapter eight explains briey the different application of Recurrent Neural Network.