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The interstellar medium (ISM) contains filamentary structure over a wide range of scales. Understanding the role of this structure, both as a conduit of gas across the scales and a diagnostic tool of local physics, is a major focus of star formation studies. We review recent progress in studying filamentary structure in the ISM, interpreting its properties in terms of physical processes, and exploring formation and evolution scenarios. We include structures from galactic-scale filaments to tenth-of-a-parsec scale filaments, comprising both molecular and atomic structures, from both observational and theoretical perspectives. In addition to the literature overview, we assemble a large amount of catalog data from different surveys and provide the most comprehensive census of filamentary structures to date. Our census consists of 22 803 filamentary structures, facilitating a holistic perspective and new insights. We use our census to conduct a meta-analysis, leading to a description of filament properties over four orders of magnitudes in length and eight in mass. Our analysis emphasize the hierarchical and dynamical nature of filamentary structures. Filaments do not live in isolation, nor they generally resemble static structures close to equilibrium. We propose that accretion during filament formation and evolution sets some of the key scaling properties of filaments. This highlights the role of accretion during filament formation and evolution and also in setting the initial conditions for star formation. Overall, the study of filamentary structures during the past decade has been observationally driven. While great progress has been made on measuring the basic properties of filaments, our understanding of their formation and evolution is clearly lacking. In this context, we identify a number of directions and questions we consider most pressing for the field.