That the K-S test works is some cute math by the father of modern probability A. N. Kolmogorov.

But that test and many more are part of non-parametric, that is, distribution-free hypothesis testing. That statistics has long been popular in the social sciences. A major theme in such statistics is permutations. Another major theme, and more recent, is resampling.

I first learned about such tests from a book that was sitting around the office, Sidney Siegel, Nonparametric Statistics for the Behavioral Sciences.

These tests are all one dimensional. Once I published a paper on a distribution-free test that is multi-dimensional -- my paper may remain the only such.

These days, see also the work of B. Efron and P. Diaconis.

More can be done.

I know of one book: Motulsky's Intuitive Biostatistics, very little formulas, much focus on upsides and downsides of various stats approaches, opinions etc.

https://speakerdeck.com/jakevdp/statistics-for-hackers is great!

HN discussion: https://news.ycombinator.com/item?id=10244950