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In order to upgrade existing electronic technology, we need simultaneously to advance power supply devices to match emerging requirements. Owing to the rapidly growing wearable and portable electronics markets, the demand to develop flexible energy storage devices is among the top priorities for humankind. Flexible supercapacitors (FSCs) have attracted tremendous attention, owing to their unrivaled electrochemical performances, long cyclability and mechanical flexibility. Carbon nanotubes (CNTs), long recognized for their mechanical toughness, with an elastic strain limit of up to 20%, are regarded as potential candidates for FSC electrodes. Along with excellent mechanical properties, high electrical conductivity, and large surface area, their assemblage adaptability from one-dimensional fibers to two-dimensional films to three-dimensional sponges makes CNTs attractive. In this review, we have summarized various assemblies of CNT structures, and their involvement in various device configurations of FSCs. Furthermore, to present a clear scenario of recent developments, we discuss the electrochemical performance of fabricated flexible devices of different CNT structures and their composites, including additional properties such as compressibility and stretchability. Additionally, the drawbacks and benefits of the study and further potential scopes are distinctly emphasized for future researchers.