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 : Ko

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 http://www.lirmm.fr/~ashen/kolmbook-eng.pdf) 2. Downey, Hirschfeldt: Algorithmic Randomness and Complexity (http://www.springer.com/gp/book/9780387955674 http://www.springer.com/gp/book/9780387955674) 3. Andre Nies: Computability and Randomness (https://global.oup.com/academic/product/computability-and-randomness-9780199230761?cc=in&lang=en& https://global.oup.com/academic/product/computability-and-ra...) in addition to the now classic book by Li and Vitanyi that others have mentioned.