Semantics is the meaning of symbols, notations, concepts, functions, and behaviors, as well as their relations that can be deduced onto a set of predefined entities and/or known concepts. Semantic computing is an emerging computational methodology that models and implements computational structures and behaviors at semantic or knowledge level beyond that of symbolic data. In semantic computing, formal semantics can be classified into the categories of to be, to have, and to do semantics. This paper presents a comprehensive survey of formal and cognitive semantics for semantic computing in the fields of computational linguistics, software science, computational intelligence, cognitive computing, and denotational mathematics. A set of novel formal semantics, such as deductive semantics, concept-algebra-based semantics, and visual semantics, is introduced that forms a theoretical and cognitive foundation for semantic computing. Applications of formal semantics in semantic computing are presented in case studies on semantic cognition of natural languages, semantic analyses of computing behaviors, behavioral semantics of human cognitive processes, and visual semantic algebra for image and visual object manipulations.