Neural networks for optimal estimation (including prediction) and/or control involve an execution step and a learning step, and are characterized by the learning step being performed by neural computations. The set of learning rules cause the circuit's connection strengths to learn to approximate the optimal estimation and/or control function that minimizes estimation error and/or a measure of control cost. The classical Kalman filter and the classical Kalman optimal controller are important examples of such an optimal estimation and/or control function. The circuit uses only a stream of noisy measurements to infer relevant properties of the external dynamical system, learn the optimal estimation and/or control function, and apply its learning of this optimal function to input data streams in an online manner. In this way, the circuit simultaneously learns and generates estimates and/or control output signals that are optimal, given the network's current state of learning.