Disclosed replacement heart valves can be designed to be delivered to a native valve site while crimped on a delivery catheter. The crimped profile of the replacement valve can be minimized by, for example, separating a frame or stent structure from a leaflet structure, along the axial direction. Disclosed replacement valves can be transitioned from a delivery configuration, in which the crimped profile can be minimized, to an operating configuration. The replacement valve can be fully assembled in both the delivery and operating configurations. In some embodiments, the leaflets can be positioned outside of the stent in the delivery configuration, and positioned inside of the stent lumen in the operating configuration. Disclosed replacement valves can include a flexible sleeve coupling the leaflets to the stent and facilitating the transition to the operating configuration. Methods of implanting said replacement valves are also disclosed.