Gamut mapping is the problem of transforming the colors of image or video content so as to fully exploit the color palette of the display device where the content will be shown, while preserving the artistic intent of the original content's creator. In particular, in the cinema industry, the rapid advancement in display technologies has created a pressing need to develop automatic and fast gamut mapping algorithms. In this article, we propose a novel framework that is based on vision science models, performs both gamut reduction and gamut extension, is of low computational complexity, produces results that are free from artifacts and outperforms state-of-the-art methods according to psychophysical tests. Our experiments also highlight the limitations of existing objective metrics for the gamut mapping problem.