Microfluidic devices with integrated membranes that enable control of mass transport in constrained environments have shown considerable growth over the last decade. Membranes are a key component in several industrial processes such as chemical, pharmaceutical, biotechnological, food, and metallurgy separation processes as well as waste management applications, allowing for modular and compact systems. Moreover, the miniaturization of a process through microfluidic devices leads to process intensification together with reagents, waste and cost reduction, and energy and space savings. The combination of membrane technology and microfluidic devices allows therefore magnification of their respective advantages, providing more valuable solutions not only for industrial processes but also for reproducing biological processes. This review focuses on membrane-based microfluidic devices for biomedical science with an emphasis on microfluidic artificial organs and organs-on-chip. We provide the basic concepts of membrane technology and the laws governing mass transport. The role of the membrane in biomedical microfluidic devices, along with the required properties, available materials, and current challenges are summarized. We believe that the present review may be a starting point and a resource for researchers who aim to replicate a biological phenomenon on-chip by applying membrane technology, for moving forward the biomedical applications.