Abstarct: The goal of this thesis is to analyze the stability of Fractional Order Echo State Neural Networks (FO-ESNN) with different properties, including time delay, memristive element in real, complex and quaternion spaces and its application in time series prediction. Besides, chaos behavior of such NN is discussed. First, the Mittag-Leffler stability of the NN equilibrium point is studied. Besides, the chaos phenomenon of this type of NN is discussed. Then, the uniform stability of the equilibrium point of such NN with constant and multiple time delays is investigated. In the follow-up, a gradient baxsed-adaptive law is proposed for optimizing the FO-ESNN parameters’, which is applied for time series prediction in stock market. Finally, the (robust) asymptotic stability of unique equilibrium point of the FO quaternion- and complex-valued ESNN with time delays and memristive elements is studied. In each part, simulation results with some numerical examples are provided to demonstrate the effectiveness of the theoretical results.