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This is a draft of the textbook/monograph that presents computability theoryusing string diagrams. The introductory chapters have been taught as graduateand undergraduate courses and evolved through 8 years of lecture notes. Thelater chapters contain new ideas and results about categorical computabilityand some first steps into computable category theory. The underlyingcategorical view of computation is based on monoidal categories with programevaluators, called *monoidal computers*. This categorical structure can beviewed as a single-instruction diagrammatic programming language called Run,whose only instruction is called RUN. This version: improved text, moved thefinal chapter to the next volume. (The final version will continue lots ofexercises and workouts, but already this version has severely degraded graphicsto meet the size bounds.)

Programs as Diagrams: From Categorical Computability to Computable Categories