A comprehensive review of the rheology and related phenomena of high internal phase ratio emulsions (referred to as HIPREs) and foams is presented. Emulsions and foams with Brownian and non-Brownian inclusions (droplets/bubbles) are considered. The topics covered are osmotic pressure, modelling and experiments of the rheology of HIPREs/foams, time-dependent rheology (thixotropy/rheopexy), normal stresses, shear banding and slip effects in flow of HIPREs/foams, influence of solid particle stabilizers (Pickering emulsion/foam), and finally pipe rheology and flow of HIPREs/foams. This is the first review article that covers all aspects of the rheology of HIPREs/foams. The theoretical and empirical models describing the osmotic pressure and rheology (yield stress, storage modulus, viscosity, etc.) of HIPREs/foams are presented and their limitations pointed out. The contributions of entropic effects in the rheology of HIPREs/foams consisting of Brownian inclusions (droplets/bubbles) are given special consideration. The key experimental studies available in the literature are reviewed including measurements of yield stress, storage modulus, and viscosity of HIPREs/foams. Comparisons of experimental data with the theoretical and semi-theoretical models are made and the limitations of the models are identified. Experimental studies elaborating special effects in HIPREs/foams rheology such as thixotropy, rheopexy, normal stresses in fixed shear strain and steady shear, shear banding in thixotropic HIPREs/foams, and slip effects are also reviewed. The effects of average size and size distribution of inclusions (droplets/bubbles) on the rheology of HIPREs/foams are evaluated. The rheology of Pickering HIPREs/foams stabilized with solid nanoparticles at the interface is reviewed and compared with the rheology of surfactant-stabilized systems. Finally, the experimental work published on the pipe flow of HIPREs/foams and its connection to rheology is presented and discussed. The gaps in the existing knowledge of the rheology of HIPREs/foams are identified and future research directions in the area are given.