Is there a natural geometric generalization of the winding number to higher dimensions?

I know it primarily as an important and useful index for closed, plane curves (e.g., the Jordan Curve Theorem), and for its role in Cauchy's theorem integrating holomorphic functions. I would be interested to learn of generalizations that essentially replace the role of the circle $\mathbb{S}^1$ with $\mathbb{S}^n$.

I've encountered references to the Fredholm index, the Pontryagin index, and to Bott periodicity, but none seem to be straightforward geometric generalizations of winding number.

This is an entirely naive question, and references and high-level descriptions would be appreciated, and more than suffice.