microRNAs (miRNAs) are important gene regulators. They control a wide range of biological processes and are involved in several types of cancers. Thus, exploring miRNA functions is important for diagnostics and therapeutics. To date, there are few feasible experimental techniques for discovering miRNA regulatory mechanisms. Alternatively, predictions of miRNA-mRNA regulatory relationships by computational methods have increasingly achieved promising results. Computational approaches are proving their ability as effective tools in reducing the number of biological experiments that must be conducted and to assist with the design of the experiments. In this review, we categorize and review different computational approaches to identify miRNA activities and functions, including the co-regulation of miRNAs and transcription factors. Our main focuses are on the recent approaches that use multiple data types for exploring miRNA functions. We discuss the remaining challenges in the evaluation and selection of models based on the results from a case study. Finally, we analyse the remaining challenges of each computational approach and suggest some future research directions.