Discrete, logic-based models are increasingly used to describe biological mechanisms. Initially introduced to study gene regulation, these models evolved to cover various molecular mechanisms, such as signalling, transcription factor cooperativity, and even metabolic processes. The abstract nature and amenability of discrete models to robust mathematical analyses make them appropriate for addressing a wide range of complex biological problems. Recent technological breakthroughs have generated a wealth of high throughput data. Novel, literature-based representations of biological processes and emerging machine learning algorithms offer new opportunities for model construction. Here, we review recent efforts to incorporate omic data into logic-based models and discuss critical challenges in constructing and analysing integrative, large-scale, logic-based models of biological mechanisms.