We survey new opportunities to design data systems, data structures and algorithms that can adapt to both data and query workloads. Data keeps growing, hardware keeps changing and new applications appear ever more frequently. One size does not fit all, but data-intensive applications would like to balance and control memory requirements, read costs, write costs, as well as monetary costs on the cloud. This calls for tailored data systems, storage, and computation solutions that match the exact requirements of the scenario at hand. Such systems should be ``synthesized'' quickly and nearly automatically, removing the human system designers and administrators from the loop as much as possible to keep up with the quick evolution of applications and workloads. In addition, such systems should ``learn'' from both past and current system performance and workload patterns to keep adapting their design.
We survey new trends in 1) self-designed, and 2) learned data systems and how these technologies can apply to relational, NoSQL, and big data systems as well as to broad data science applications. We focus on both recent research advances and practical applications of this technology, as well as numerous open research opportunities that come from their fusion. We specifically highlight recent work on data structures, algorithms, and query optimization, and how machine learning inspired designs as well as a detailed mapping of the possible design space of solutions can drive innovation to create tailored systems. We also position and connect with past seminal system designs and research in auto-tuning, modular/extensible, and adaptive data systems to highlight the new challenges as well as the opportunities to combine past and new technologies.