Software as a Service (SaaS) is widely used and depended on by a wide range of applications. Considering this, the SaaS should capacitate itself to offer service to a large number of customers having their own specific requirements, without encountering software quality problems. Therefore, several researchers delved into the SaaS customization, and many customization solutions have been proposed. However, heretofore, no analysis or study explicitly classifies these proposals using different criteria, e.g., the kind of change required, the component of the software requiring changes, and the quality attributes of the SaaS considered in each proposed solution. This paper adopts the systematic mapping approach to methodically investigate the solutions recommended for the SaaS customization problems. These solutions are classified into various categories to create a classification scheme based on the customization types (personalization, configuration, composition, modification, integration, and extension), customization layer (user interfaces, workflows, services, and data), and quality attributes. Our study identified 81 primary studies reporting SaaS customization solutions. The results show that the configuration, composition, and extension received the highest consideration in the proposed solutions. In addition, the majority of the proposed solutions for the SaaS customization are connected with the workflow and service layers. Furthermore, the attributes, such as multi-tenancy, security, functionality, scalability, availability, and efficiency, are considered much more often than other attributes. The classification of the proposed solutions for the SaaS customization and results of this paper can play an important role in creating a framework for the SaaS customization assessment.