The real-time recurrent learning algorithm is a gradient-following learning algorithm for completely recurrent networks running in continually sampled time. Here we use a series of simulation experiments to investigate the power and properties of this algorithm. In the recurrent networks studied here, any unit can be connected to any other, and any unit can receive external input. These networks run continually in the sense that they sample their inputs on every update cycle, and any unit can have a training target on any cycle. The storage required and computation time on each step are independent of time and are completely determined by the size of the network, so no prior knowledge of the temporal structure of the task being learned is required. The algorithm is nonlocal in the sense that each unit must have knowledge of the complete recurrent weight matrix and error vector. The algorithm is computationally intensive in sequential computers, requiring a storage capacity of the order of the third power of the number of units and a computation time on each cycle of the order of the fourth power of the number of units. The simulations include examples in which networks are taught tasks not possible with tapped delay lines—that is, tasks that require the preservation of state over potentially unbounded periods of time. The most complex example of this kind is learning to emulate a Turing machine that does a parenthesis balancing problem. Examples are also given of networks that do feedforward computations with unknown delays, requiring them to organize into networks with the correct number of layers. Finally, examples are given in which networks are trained to oscillate in various ways, including sinusoidal oscillation.