Miniaturization has been the driving force of scientific and technological advances over recent decades. Recently, flexibility has gained significant interest, particularly in miniaturization approaches for biomedical devices, wearable sensing technologies, and drug delivery. Flexible microfluidics is an emerging area that impacts upon a range of research areas including chemistry, electronics, biology, and medicine. Various materials with flexibility and stretchability have been used in flexible microfluidics. Flexible microchannels allow for strong fluid-structure interactions. Thus, they behave in a different way from rigid microchannels with fluid passing through them. This unique behaviour introduces new characteristics that can be deployed in microfluidic applications and functions such as valving, pumping, mixing, and separation. To date, a specialised review of flexible microfluidics that considers both the fundamentals and applications is missing in the literature. This review aims to provide a comprehensive summary including: (i) Materials used for fabrication of flexible microfluidics, (ii) basics and roles of flexibility on microfluidic functions, (iii) applications of flexible microfluidics in wearable electronics and biology, and (iv) future perspectives of flexible microfluidics. The review provides researchers and engineers with an extensive and updated understanding of the principles and applications of flexible microfluidics.