Medium- and long-chain alkanes and their mixtures possess a remarkable physical property - they form intermediate structured phases between their isotropic liquid phase and their fully ordered crystal phase. These intermediate phases are called "rotator phases" or "plastic phases" (soft solids) because the incorporated alkane molecules possess a long-range positional order while preserving certain mobility to rotate, which results in complex visco-plastic rheological behaviour. The current article presents a brief overview of our current understanding of the main phenomena involved in the formation of rotator phases from single alkanes and their mixtures. In bulk, five rotator phases with different structures were identified and studied in detail. Along with the thermodynamically stable rotator phases, metastable and transient (short living) rotator phases were observed. Bulk rotator phases provided important information about several interfacial phenomena of high scientific interest, such as the energy of crystal nucleation, entropy and enthalpy of alkane freezing, interfacial energy between a crystal and its melt, etc. In alkane mixtures, the region of existence of rotator phases increases significantly, reflecting the disturbed packing of different molecules. All these phenomena are very important in the context of alkane applications as lubricants, in cosmetics, as phase-change materials for energy storage, etc. Significant expansion of the domain of rotator phases was observed also in confinements - in the pores of solid materials impregnated with alkanes, in polymeric microcapsules containing alkanes, and in micrometer sized emulsion droplets. The rotator phases were invoked to explain the mechanisms of two recently discovered phenomena in cooled alkane-in-water emulsions - the spontaneous "self-shaping" and the spontaneous "self-bursting" (fragmentation) of emulsion drops. The so-called "α-phases" formed by fatty acids and alcohols, and the "gel phase" formed in phospholipid and soap systems exhibit structural characteristics similar to those in the alkane rotator phases. The subtle connections between all these diverse systems are outlined, providing a unified outlook of the main phenomena related to the formation of such soft solid materials. The occurrence of alkane rotator phases in natural materials and in several technological applications is also reviewed to illustrate the general importance of these unique materials and the related phenomena.