Supercapacitors are the latest development in the field of energy storage devices (ESDs). A lot of research has been done in the last few decades to increase the performance of supercapacitors. The electrodes of supercapacitors are modified by composite materials based on conducting polymers, metal oxide nanoparticles, metal-organic frameworks, covalent organic frameworks, MXenes, chalcogenides, carbon nanotubes (CNTs), etc. In comparison to rechargeable batteries, supercapacitors have advantages such as quick charging and high power density. This review is focused on the progress in the development of electrode materials for supercapacitors using composite materials based on conducting polymers, graphene, metal oxide nanoparticles/nanofibres, and CNTs. Moreover, we investigated different types of ESDs as well as their electrochemical energy storage mechanisms and kinetic aspects. We have also discussed the classification of different types of SCs; advantages and drawbacks of SCs and other ESDs; and the use of nanofibres, carbon, CNTs, graphene, metal oxide-nanofibres, and conducting polymers as electrode materials for SCs. Furthermore, modifications in the development of different types of SCs such as pseudo-capacitors, hybrid capacitors, and electrical double-layer capacitors are discussed in detail; both electrolyte-based and electrolyte-free supercapacitors are taken into consideration. This review will help in designing and fabricating high-performance supercapacitors with high energy density and power output, which will act as an alternative to Li-ion batteries in the future.