The accurate identification and classification of local ordered and disordered structures is an important task in atomistic computer simulations. Here, we demonstrate that properly trained artificial neural networks can be used for this purpose. Based on a neural network approach recently developed for the calculation of energies and forces, the proposed method recognizes local atomic arrangements from a set of symmetry functions that characterize the environment around a given atom. The algorithm is simple and flexible and it does not rely on the definition of a reference frame. Using the Lennard-Jones system as well as liquid water and ice as illustrative examples, we show that the neural networks developed here detect amorphous and crystalline structures with high accuracy even in the case of complex atomic arrangements, for which conventional structure detection approaches are unreliable.