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Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem -- e.g., factor graphs -- neural networks are usually monolithic mappings from inputs to outputs, with a fixed computation order. This limitation prevents them from capturing different directions of computation and interaction between the modeled variables. In this paper, we combine the representational strengths of factor graphs and of neural networks, proposing undirected neural networks (UNNs): a flexible framework for specifying computations that can be performed in any order. For particular choices, our proposed models subsume and extend many existing architectures: feed-forward, recurrent, self-attention networks, auto-encoders, and networks with implicit layers. We demonstrate the effectiveness of undirected neural architectures, both unstructured and structured, on a range of tasks: tree-constrained dependency parsing, convolutional image classification, and sequence completion with attention. By varying the computation order, we show how a single UNN can be used both as a classifier and a prototype generator, and how it can fill in missing parts of an input sequence, making them a promising field for further research.