Work-related Musculoskeletal Disorders (WMSDs) are the most common occupational diseases caused by the prolonged performance of strenuous work, such as manual handling of loads or long-term maintenance of incongruous postures. Different safety protocols are implemented to reduce WMSDs and optimize the working environment, but one of the most promising solutions is using occupational exoskeletons (OEs). However, to truly acknowledge the benefits of OEs and be able to introduce them into daily business use, devices must pass several development and testing stages that determine the Technology Readiness Level (TRL). This review study aims to present an up-to-date collection of the most advanced assessments of exoskeletons for upper and back support, ranging from laboratory real-task simulations to operational scenarios in industrial sites. To identify relevant studies, we conducted comprehensive searches across different electronic databases, i.e., PubMed, Scopus, and Web of Science. Different keywords were used for the literature search, e.g., occupational exoskeleton, industrial exoskeleton, etc. Studies were included if they investigated the assessment of exoskeletons in the laboratory with real tasks or an industrial environment. We identified 45 research articles that fulfilled this selection criterion. Several features are compared and discussed in detail, such as industrial environment, experimental protocol, task performed, and exoskeleton typology. These data allowed us to formulate results that report the correspondence or discrepancy between the number of papers testing exoskeletons and WMSDs in different industrial sectors, the type of assessment performed, and the impact of exoskeletons on workers and industries at different TRLs. Among the results, the incidence of WMSDs in the manufacturing industry is 21.13%, while the adoption of exoskeletons in the same field is the highest with respect to the other industrial fields, at 44.45%. Electromyography (EMG) and Questionnaires were the most evaluated typologies across all development and testing stages (with an incidence of 64% across the selected articles). Additionally, an average reduction of EMG activity was reported, with 24% for Upper Limb and 20% for Back Support. Regarding the subjective assessment reported in the questionnaires, 68% of the studies reported a positive evaluation. Based on these outcomes, this work provides a framework for an effective evaluation process for the OEs to raise TRL with recommendations for future research activities.