Filtration of molecules by nanometer-sized structures is ubiquitous in our everyday life, but our understanding of such molecular filtration processes is far less than desired. Until recently, one of the main reasons was the lack of experimental methods that can help provide detailed, microscopic pictures of molecule-nanostructure interactions. Several innovations in experimental methods, such as nuclear track-etched membranes developed in the 70s, and more recent development of nanofluidic molecular filters, played pivotal roles in advancing our understanding. With the ability to make truly molecular-scale filters and pores with well-defined sizes, shapes, and surface properties, now we are well positioned to engineer better functionality in molecular sieving, separation and other membrane applications. Reviewing past theoretical developments (often scattered across different fields) and connecting them to the most recent advances in the field would be essential to get a full, unified view on this important engineering question.