Modern database systems provide a general design principle for various data types and application workloads. While gaining great success in the last decades, the principle has a limitation that a database system may not achieve superior performance, if the system cannot be "customized" to the specific data distributions and workload characteristics. To address the problem, learnable database systems have attracted much attention from both industrial and academic communities, with a novel idea of using machine learning to optimize database systems. Along with this direction, extensive efforts have been done very recently to advance the field of learnable database systems. This survey systematically reviews the existing studies from the perspective of database system architecture. A fine-grained taxonomy is provided by categorizing the existing works by their target learnable database components. To help readers better understand each type of learnable components their motivations are presented, demonstrating the insights and introducing the key techniques. Finally, a number of promising future research directions are outlined of learnable database systems.