Lab-on-a-chip (LOC) and organ-on-a-chip (OOC) devices are highly versatile platforms that enable miniaturization and advanced controlled laboratory functions (i.e., microfluidics, advanced optical or electrical recordings, high-throughput screening). The manufacturing advancements of LOCs/OOCs for biomedical applications and their current limitations are briefly discussed. Multiple studies have exploited the advantages of mimicking organs or tissues on a chip. Among these, we focused our attention on the brain-on-a-chip, blood–brain barrier (BBB)-on-a-chip, and neurovascular unit (NVU)-on-a-chip applications. Mainly, we review the latest developments of brain-on-a-chip, BBB-on-a-chip, and NVU-on-a-chip devices and their use as testing platforms for high-throughput pharmacological screening. In particular, we analyze the most important contributions of these studies in the field of neurodegenerative diseases and their relevance in translational personalized medicine.