Shape memory alloys (SMAs) have found widespread applications as biomedical devices. Biocompatibility, corrosion resistance, and ductility make these alloys attractive for medical devices such as stents and filters. For these implants, the superelastic property is the primary function of SMAs. Additionally, these alloys, such as NiTi as the prime example, can be used for actuation. Several modes of actuation such as displacement control, force control, and compliance control have been used as harnesses with SMA devices. These two unique properties have opened another application in the form of neurosurgery and robot-assisted surgery devices, as well as controlled assistive and rehabilitation devices. This paper reviews the state of the art of application of SMAs in the latter category where control is applied to harness innovative medical devices. To this end, two major subsets of these devices: prosthesis and orthosis which take the advantage of SMAs in assistive and rehabilitation devices are studied. These devices are further categorized to hand prosthetics, elbow, knee and ankle orthotics. In most of these designs, SMA wires act as artificial muscles to mimic the motion of limbs in the target joints. The evolution of each category is explained, and the specific results of them are reported. The paper also reviews the SMA applications for neurological and neuromuscular rehabilitation. To this end, different categories of rehabilitation devices as a passive and aided exercise for the ankle, knee, and elbow are highlighted. The SMA actuator in these devices can be EMG-controlled to improved patient outcome. In addition to providing a comprehensive overview of the biomedical devices, this paper identifies several possible future directions of SMA related research in the area of assistive and rehabilitation devices.