Choosing a suitable topology for a neural network, given an application, is a difficult problem. Usually, after a tedious trial-and-error process, an oversized topology is chosen, which is prone to various drawbacks like a high demand on computational resources and a high generalization error. A way to solve this is to trim the network size during the training process. This is done with so-called \emph{pruning} methods, of which an overview is given. From these methods, those that are potentially suitable for high order perceptrons are selected, and then adapted accordingly. Next, they are tested on a variety of benchmarks by means of a large number of experiments. The conclusions are both of a generic nature, pointing out some pitfalls of neural network pruning in general, and of a more specific nature, identifying the best pruning methods for high order perceptrons.