Filamentous structures are ubiquitous in nature, are studied in diverse scientific fields, and span vastly different spatial scales. Filamentous structures in biological systems fulfill different functions and often form dynamic networks that respond to perturbations. Therefore, characterizing the properties of filamentous structures and the networks they form is important to gain better understanding of systems level functions and dynamics. Filamentous structures are captured by various imaging technologies, and analysis of the resulting imaging data addresses two problems: (i) identification (tracing) of filamentous structures in a single snapshot and (ii) characterizing the dynamics (i.e., tracking) of filamentous structures over time. Therefore, considerable research efforts have been made in developing automated methods for tracing and tracking of filamentous structures. Here, we provide a systematic review in which we present, categorize, and discuss the state-of-the-art methods for tracing and tracking of filamentous structures in sparse and dense networks. We highlight the mathematical approaches, assumptions, and constraints particular for each method, allowing us to pinpoint outstanding challenges and offer perspectives for future research aimed at gaining better understanding of filamentous structures in biological systems.