Membranes play a key role in many organ-on-a-chip (OOC) platforms due to nutrient and mass transport, barrier formation, and mechanical signaling. However, despite their key role in OOC devices, the relationship between membrane design parameters and device performance is limited in the literature. Here, this review gives a critical overview of not only the types of membranes for OOC devices (natural, synthetic, hybrid, and emerging functional membranes) but also defines the key engineering design requirements for the selection and integration of membranes with microfluidic OOC platforms. Additionally, it presents types of organ-specific OOC devices (lung, kidney, liver, blood–brain barrier, and intestine models), highlighting existing challenges and limitations in the fabrication of such platforms. The paper also presents future directions toward smart, multifunctional membranes.