High-performance logic circuits that are constructed on flexible or unconventional substrates are required for emerging applications such as real-Time analytics. Carbon nanotube thin-film transistors (TFTs) are attractive for these applications because of their high mobility and low cost. However, flexible nanotube TFTs usually suffer from much lower performance than those built on rigid substrates, and the resulting flexible integrated circuits typically exhibit low-speed operation with logic gate delays of over 1 μ s, which severely limits their practical application. Here we show that high-performance carbon nanotube TFTs and complementary circuits can be fabricated on flexible polyimide substrates using a high-yield, scalable process. Our flexible TFTs exhibit state-of-The-Art performance with very high current densities (> 17 μ A μ m-1), large current on/off ratios (> 106), small subthreshold slopes (< 200 mV dec-1), high field-effect mobilities (-50 cm2 V-1 s-1) and excellent flexibility. We also develop a reliable n-Type doping process, which allows us to fabricate complementary logic gates and integrated circuits on flexible substrates. With our approach, we build flexible ring oscillators that have a stage delay of only 5.7 ns.