The Industry 5.0 transition highlights EU efforts to design intelligent devices that can work alongside humans to enhance human capabilities, and such vision aligns with user preferences and needs to feel safe while collaborating with such systems take priority. This demands a human-centric research vision and requires a societal and educational shift in how we perceive technological advancements. To better understand this perspective, we conducted a systematic literature review focusing on understanding how trust and trustworthiness can be key aspects of supporting this move towards Industry 5.0. This review aims to overview the most common methodologies and measurements and collect insights about barriers and facilitators for fostering trustworthy HRI. After a rigorous quality assessment following the Systematic Reviews and Meta-Analyses guidelines, using rigorous inclusion criteria and screening by at least two reviewers, 34 articles were included in the review. The findings underscores the significance of trust and safety as foundational elements for promoting secure and trustworthy human-machine cooperation. Confirm that almost 30% of the revised articles do not present a definition of trust, which can be problematic as this lack of conceptual clarity can undermine research efforts in addressing this problem from a central perspective. It highlights that the choice of domain and area of application should influence the choice of methods and approaches to fostering trust in HRI, as those choices can significantly affect user preferences and their perceptions and assessment of robot capabilities. Additionally, this lack of conceptual clarity can be a potential barrier to fostering trust in HRI and explains the sometimes contradictory findings or choice of methods and instruments used to investigate trust in robots and other autonomous systems in the literature.