In many classification and prediction problems it is known that the response variable depends on certain explanatory variables. Monotone neural networks can be used as powerful tools to build monotone models with better accuracy and lower variance compared to ordinary nonmonotone models. Monotonicity is usually obtained by putting constraints on the parameters of the network. In this paper, we will clarify some of the theoretical results on monotone neural networks with positive weights, issues that are sometimes misunderstood in the neural network literature. Furthermore, we will generalize some of the results obtained by Sill for the so-called min-max networks to the case of partially monotone problems. The method is illustrated in practical case studies.