It has remained unknown whether one can in principle carry out reliable digital computations with networks of biologically realistic models for neurons. This article presents rigorous constructions for simulating in real-time arbitrary given boolean circuits and fi(cid:173) nite automata with arbitrarily high reliability by networks of noisy spiking neurons. In addition we show that with the help of "shunting inhibition" even networks of very unreliable spiking neurons can simulate in real-time any McCulloch-Pitts neuron (or "threshold gate"), and therefore any multilayer perceptron (or "threshold circuit") in a reliable manner. These constructions provide a possible explana(cid:173) tion for the fact that biological neural systems can carry out quite complex computations within 100 msec. It turns out that the assumption that these constructions require about the shape of the EPSP's and the behaviour of the noise are surprisingly weak.