The gate dielectric layer is an essential element of field-effect transistors (FETs), large area integrated circuits, and various application electronics. Beyond basic insulation between the semiconductor layer and gate electrode, FET performance largely depends on the capacitance of the chosen insulator layer properties. Recent functional devices developments require new functionalities, such as high mechanical flexibility and stretchability in addition to basic insulating and physical properties. This review focuses on recent developments of high-capacitance polymer gate dielectric materials and their application for low-voltage flexible FETs. The fundamental principles and components of transistors are discussed briefly, then the underlying concepts for gate dielectrics and the mechanisms for polymer dielectric properties, such as capacitance control, leakage current formation in dielectric polymer films, and patternability, are described. Strategic approaches to achieve high capacitance based on low-k dielectric materials, and the development and use of high-capacitance polymers and hybrid polymer gate dielectrics in the context of emerging transistors on flexible substrates are also discussed.