What are some examples of engineering influenced by nature's "designs?" Have humans ever engineered anything they didn't understand by imitating nature?

Natural selection optimizes creatures without having to understand the mechanisms with which it does so. For example, the mechanism that causes cats to land on their feet is based on some fairly complex rotational physics, but of course the cats don’t know what they’re doing. They just do it because hundreds of thousands of cats that didn’t do it plunged to their death some thousands of years ago.

Knowing that natural selection very effectively optimizes creatures, we’d be stupid not to draw on some of its “design” concepts in our engineering work. What are some examples of our doing so? And have we ever engineered anything without fully understanding the underlying mechanics of what we’re imitating?

As a secondary question, what are some examples of behaviors or abilities that creatures display of which we don’t understand the mechanics?