Surface tension force plays a vital role when the length scale of a system is small (i.e., in the micro/nano meter range). In the case of compliant structures and soft substrates, the competition between interfacial energy and elastic strain energy in the bulk gives rise to many interesting phenomena. Elastocapillarity refers to the study of the effects of deformation of flexible structures or soft substrates under the action of capillary forces. In contrast to rigid structures/substrates, the liquid surface tension force at the three-phase contact line could deform compliant structures or soft solids, thus resulting in interesting dynamics. The elastocapillary interaction between liquid and solid can give rise to stiction in microstructures, wrapping of droplet by flexible structures, buckling of fibres, and ridge formation at the contact line of a droplet on a soft substrate. Besides, recent research has shown that elastocapillarity can aid in the manipulation of liquid flow through flexible confinements and over soft substrates. Here, we report an extensive review of the literature on elastocapillarity mainly focusing on the elastocapillary-based liquid transport. The research in the field has been broadly classified into three different categories: liquid flow in compliant microchannels, liquid flow in flexible structures, and transport of droplets over soft elastic substrates. In each of the above cases, theoretical understanding of the mechanism responsible for liquid transport, and experimental studies leading to interesting results and observations are presented and discussed. Numerical modelling of elastocapillarity phenomena is also presented. Finally, we discuss the research challenges and future directions in the field of elastocapillarity-based transport of liquids. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.