Spasticity is a motor disorder that causes stiffness or tightness of the muscles and can interfere with normal movement, speech, and gait. Traditionally, the spasticity assessment is carried out by clinicians using standardized procedures for objective evaluation. However, these procedures are manually performed and, thereby, they could be influenced by the clinician’s subjectivity or expertise. The automation of such traditional methods for spasticity evaluation is an interesting and emerging field in neurorehabilitation. One of the most promising approaches is the use of robot-aided systems. In this paper, a systematic review of systems focused on the assessment of upper limb (UL) spasticity using robotic technology is presented. A systematic search and review of related articles in the literature were conducted. The chosen works were analyzed according to the morphology of devices, the data acquisition systems, the outcome generation method, and the focus of intervention (assessment and/or training). Finally, a series of guidelines and challenges that must be considered when designing and implementing fully-automated robot-aided systems for the assessment of UL spasticity are summarized.