[sn41]

There are a lot of good recent books on this topic (I don't know if this area is undergoing a revival) :

1. N. K. Vereschagin, V. Uspensky, Alexander Shen : Kolmogorov Complexity (English draft in preparation: http://www.lirmm.fr/~ashen/kolmbook-eng.pdf)

2. Downey, Hirschfeldt: Algorithmic Randomness and Complexity (http://www.springer.com/gp/book/9780387955674)

3. Andre Nies: Computability and Randomness (https://global.oup.com/academic/product/computability-and-ra...)

in addition to the now classic book by Li and Vitanyi that others have mentioned.