The theory of shadowed sets is one among several key contributors to the area of granular computing. As the name stipulates, granular computing embraces processing of information granules. By information granules we mean collections of entities being assembled together in order to achieve a certain conceptual and/or computational feasibility of processing carried out in any complex system, no matter whether natural or artificial. Granular computing, as being exclusively geared toward processing of information granules, subsumes the commonly encountered numeric style of processing. The intent of shadowed sets is to capture ("isolate" or "localize") and quantify the factor of uncertainty inherently existing in any real-world system. We first discuss the underlying theoretical underpinnings of shadowed sets that primarily dwell on the pillar of three-valued logic. We also come up with a number of illustrative examples that help grasp the essence of the concept. The study embarks on a variety of the applications of shadowed sets to fuzzy mappings along with an analysis of their relevance as well as data quantization. In the case of fuzzy mappings, it is revealed that shadowed sets provide an interesting three-valued quantification of the property of relevancy (such as acceptable mapping, marginal mapping, and a lack of mapping). This article includes a number of detailed calculations concerning two commonly exploited classes of triangular and Gaussian fuzzy sets. Moreover, we elaborate on the exploration of shadowed sets as an algorithmic realization of the least commitment principle advocated by Marr.