Context: Machine learning (ML) is nowadays so pervasive and diffused that virtually no application can avoid its use. Nonetheless, its enormous potential is often tempered by the need to manage non-functional requirements (NFRs) and navigate pressing, contrasting trade-offs. Objective: In this respect, we notice a lack of systematic synthesis of challenges explicitly tied to achieving and managing NFRs in ML-enabled systems. Such a synthesis may not only provide a comprehensive summary of the state of the art but also drive further research on the analysis, management, and optimization of NFRs of ML-enabled systems. Method: In this paper, we propose a systematic literature review targeting two key aspects such as (1) the classification of the NFRs investigated so far, and (2) the challenges associated with achieving and managing NFRs in ML-enabled systems during model development Through the combination of well-established guidelines for conducting systematic literature reviews and additional search criteria, we survey a total amount of 130 research articles. Results: Our findings report that current research identified 31 different NFRs, which can be grouped into six main classes. We also compiled a catalog of 26 software engineering challenges, emphasizing the need for further research to systematically address, prioritize, and balance NFRs in ML-enabled systems. Conclusion: We conclude our work by distilling implications and a future outlook on the topic.