String diagrams are a powerful computational tool, most commonly used in the context of tensor categories and occasionally bicategories. I will talk about work in progress on extending this to higher categories. The idea is to define a semistrict $n$-category as something which admits composites for labeled string diagrams, much as one can define a strict $n$-category as something that admits composites for pasting diagrams. This notion of semistrict $n$-category should be more general than that of a strict $n$-category, but not as general as that of a weak $n$-category. We can show that semistrict $3$-categories are the same thing as Gray categories and it is known that every weak $3$-category (also called a tricategory) is equivalent to a Gray category. It is not known whether something similar holds in higher dimensions. I will also try to give an idea of the usefulness of string diagram calculus in dimensions $3$ and $4$, by showing how it can be used to prove coherence theorems for adjunctions.