<p><b>Abstract</b>—This paper discusses the ability of multilayer perceptrons (MLPs) to model the probability distribution of data in typical pattern recognition and verification problems. It is proven that multilayer perceptrons with sigmoidal units and a number of hidden units less or equal than the number of inputs are unable to model patterns distributed in typical clusters, since these networks draw open separation surfaces in the pattern space. When using more hidden units than inputs, the separation surfaces can be closed but, unfortunately, it is proven that determining whether or not an MLP draws closed separation surfaces in the pattern space is <tmath>${\cal NP}$</tmath>-hard. The major conclusion of this paper is somewhat opposite to what is believed and reported in many application papers: MLPs are definitely not adequate for applications of pattern recognition requiring a reliable rejection and, especially, they are not adequate for pattern verification tasks.</p>