The ability to identify and characterize not only the protein-protein interactions but also their internal modular organization through network analysis is fundamental for understanding the mechanisms of biological processes at the molecular level. Indeed, the detection of the network communities can enhance our understanding of the molecular basis of disease pathology, and promote drug discovery and disease treatment in personalized medicine. This work gives an overview of recent computational methods for the detection of protein complexes and functional modules in protein-protein interaction networks, also providing a focus on some of its applications. We propose a systematic reformulation of frequently adopted taxonomies for these methods, also proposing new categories to keep up with the most recent research. We review the literature of the last five years (2017-2021) and provide links to existing data and software resources. Finally, we survey recent works exploiting module identification and analysis, in the context of a variety of disease processes for biomarker identification and therapeutic target detection. Our review provides the interested reader with an up-to-date and self-contained view of the existing research, with links to state-of-the-art literature and resources, as well as hints on open issues and future research directions in complex detection and its applications.