Learning from demonstration is one of the most promising methods to counteract the challenging long-term trends in repetitive industrial assembly. It offers not only a programming technique that is accessible to workers on the shop floor, reducing the need for robot experts and the associated costs but also a possible solution to the observable shift from mass-production to mass-customisation through flexible and generalising systems. Since the emergence of the learning from demonstration idea in the 1980s, its methodologies, capabilities, and achievements have constantly evolved. However, despite reports of continued progress in academic publications, the concept has not yet robustly emerged across the assembly industry. In light of its great potential, this paper presents the findings from a systematic literature review following the updated Preferred Reporting Items for Systematic Reviews (PRISMA) guidelines. It aims to provide an overview of the state-of-the-art learning from demonstration solutions developed for assembly-related tasks and offer a critical discussion of remaining obstacles in order to drive its progression towards meaningful deployments. The analysis includes a total of 61 papers over the period of 2013-2023 sourced from Scopus and Web of Science databases. Findings indicate that learning from demonstration has attained a significant level of maturity within the research environment, as evidenced by thorough experimental achievements, proving its great promise for industrial assembly applications. However, critical obstacles exist in the area of proven practicability, task complexity and diversity, generalisation, performance evaluation and integration concepts that require attention to promote its widespread adoption and create a seamless transition into industrial practices.