Solid oxide fuel cells (SOFC) have emerged as energy conversion devices in achieving high efficiency of over 70% with regeneration. The critical components of SOFC include anode, electrolyte, and cathode. However, for a stack of individual SOFCs, the evaluation of sealants and interconnects are also becoming essential. In this review article, material selection, fundamentals of operation and underlying mechanisms, processing, microstructural and phase characterization, and the functionality and performance of individual SOFC components are presented in details. The major challenges and complexity in functional section of SOFC includes: (i) poisoning via sulphur and coke deposition, surface diffusion of adsorbate, and charge transfer at triple-phase boundary (TPB) in anode, (ii) hindered O2− migration that converts chemical energy into electrical energy in the solid electrolyte (thus, the creation of ion transfer channels, ease of O2− migration, dissociation of vacancy around dopants, straining of lattice, and other factors such as control of phase and its distribution, grain and grain boundary conductivity, etc., becomes critical in designing the electrolytes for SOFCs), (iii) multiple rate determining factors such as geometry of active surfaces, existence of overpotential, etc., in cathode (thereby, comprehensive electrochemical impedance spectroscopy is required for the analysis of solid cathodes in SOFC), (iv) chemical incompatibility and instability in both oxidizing and reducing environments while matching the coefficient of thermal expansion (CTE) in the interconnects in order to sustain large number of thermal cycling during the operation of SOFC, and (v) isolation of the fuel and oxidizing gases while matching the CTE of the anode, cathode and interconnects, using sealant. Moreover, the glass-transition of sealant dictates the allowable maximum working operation temperature of SOFC. Thus, the necessitated temporal progress in material selection along with a detailed insight into the conceptual role of thermodynamics and kinetics of surface/cell reactions, effect of phases and microstructure on conductivity, fuel flexibility and deterioration in performance of individual fuel cell components, and evolution of new materials are coherently presented. Therefore, this article provides a comprehensive review with respect to the structure, chemistry, design and selection of materials, underlying mechanisms, and performance of each SOFC component, and it opens up the future directions towards pursuing SOFC research.