For applications in sensing and cavity-based quantum computing and metrology, open-access Fabry-Perot cavities – with an air or vacuum gap between a pair of high reflectance mirrors – offer important advantages compared to other types of microcavities. For example, they are inherently tunable using MEMS-based actuation strategies, and they enable atomic emitters or target analytes to be located at high field regions of the optical mode. Integration of curved-mirror Fabry-Perot cavities on chips containing electronic, optoelectronic, and optomechanical elements is a topic of emerging importance. Micro-fabrication techniques can be used to create mirrors with small radius-of-curvature, which is a prerequisite for cavities to support stable, small-volume modes. We review recent progress towards chip-based implementation of such cavities, and highlight their potential to address applications in sensing and cavity quantum electrodynamics.