Where can I read reviews of mathematical theories?

On the popular level, there is an AMS feature column: http://www.ams.org/samplings/feature-column/fc-current.cgi and a series of 10 volumes "What's happening in mathematical sciences": https://bookstore.ams.org/HAPPENING, also by AMS.

On a higher level, there are journals which publish surveys, addressed to the general audience of mathematicians: Bulletin of the AMS, Russian Math Surveys, Sugaku expositions, Gazette des Mathematiciens, Expositiones Mathematicae, L’Enseignement Mathématique, and several other such journals.

An addendum to Alexandre's answer: I have noticed that in some "proceedings" or "conference" volumes, the first article (or the introduction) is a very nice overview of the theory that is treated in the book. The same can be found in books with the title "on the occasion of the [age]-th birthday of [some big name in mathematics]".