Research on deep learning-powered voice conversion (VC) in speech-to-speech scenarios is getting increasingly popular. Although many of the works in the field of voice conversion share a common global pipeline, there is a considerable diversity in the underlying structures, methods, and neural sub-blocks used across research efforts. Thus, obtaining a comprehensive understanding of the reasons behind the choice of the different methods in the voice conversion pipeline can be challenging, and the actual hurdles in the proposed solutions are often unclear. To shed light on these aspects, this paper presents a scoping review that explores the use of deep learning in speech analysis, synthesis, and disentangled speech representation learning within modern voice conversion systems. We screened 621 publications from more than 38 different venues between the years 2017 and 2023, followed by an in-depth review of a final database consisting of 123 eligible studies. Based on the review, we summarise the most frequently used approaches to voice conversion based on deep learning and highlight common pitfalls within the community. Lastly, we condense the knowledge gathered, identify main challenges and provide recommendations for future research directions.