We introduce monoidal width as a measure of the difficulty of decomposing morphisms in monoidal categories. By instantiating monoidal width and two variations in a suitable category of cospans of graphs, we capture existing notions, namely branch width, tree width and path width. By changing the category of graphs, we are also able to capture rank width. Through these examples, we propose that monoidal width: (i) is a promising concept that, while capturing known measures, can similarly be instantiated in other settings, avoiding the need for ad-hoc domain-specific definitions and (ii) comes with a general, formal algebraic notion of decomposition using the language of monoidal categories.