Background: As Machine Learning (ML) advances rapidly in many fields, it is being adopted by academics and businesses alike. However, ML has a number of different challenges in terms of maintenance not found in traditional software projects. Identifying what causes these maintainability challenges can help mitigate them early and continue delivering value in the long run without degrading ML performance. Aim: This study aims to identify and synthesise the maintainability challenges in different stages of the ML workflow and understand how these stages are interdependent and impact each other's maintainability. Method: Using a systematic literature review, we screened more than 13000 papers, then selected and qualitatively analysed 56 of them. Results: (i) a catalogue of maintainability challenges in different stages of Data Engineering, Model Engineering workflows and the current challenges when building ML systems are discussed; (ii) a map of 13 maintainability challenges to different interdependent stages of ML that impact the overall workflow; (iii) Provided insights to developers of ML tools and researchers. Conclusions: In this study, practitioners and organisations will learn about maintainability challenges and their impact at different stages of ML workflow. This will enable them to avoid pitfalls and help to build a maintainable ML system. The implications and challenges will also serve as a basis for future research to strengthen our understanding of the ML system's maintainability.