This paper starts a brief historical overview of occurrence and development of fuzzy systems and their applications. Integration methods are proposed to construct a fuzzy system using other AI methods, achieving synergy effect. Accuracy and interpretability are selected as main properties of rule-based fuzzy systems. The tradeoff between interpretability and accuracy is considered to be the actual problem. The purpose of this paper is the in-depth study of the methods and tools to achieve a tradeoff for accuracy and interpretability in rule-based fuzzy systems and to describe our interpretability indexes. A comparison of the existing ways of interpretability estimation has been made We also propose the new way to construct heuristic interpretability indexes as a quantitative measure of interpretability. In the main part of this paper we describe previously used approaches, the current state and original authors’ methods for achieving tradeoff between accuracy and complexity.