Compact object binaries formed from dynamical interactions will generically have non-zero orbital eccentricity. The gravitational waves from such binaries can change drastically depending on how large the eccentricity is, ranging from emitted in small a subset of orbital harmonics at low eccentricities, to being concentrated into intense bursts of radiation from each pericenter passage at large eccentricities. Gravitational waves from such highly eccentric binaries present themselves an intriguing systems for probing fundamental physics, but also present interesting challenges in terms of detection. The presence of orbital eccentricity in the gravitational wave signature gives an unequivocal method of determining the origin of the binary, while the highly dynamical nature of pericenter passage often enhances the physics associated with matter and gravity. The generation of faithful models of the repeated burst signals has proven challenging, and often sub-optimal detection strategies like power stacking are considered for detection. This chapter reviews our current understanding of eccentric binaries, the leading efforts to model their gravitational wave emission, and the physics that can be probed with these detections.