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I would like to ask suggestions for good texts which introduce circuit complexity. Any pointers to recent advances and open problems in this field would also be helpful.

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To start off, there is of course Arora and Barak's book Computational Complexity: A Modern Approach. From there, parts 3 and 4 of Jukna's book Boolean Function Complexity: Advances and Frontiers make excellent reading material. Also, Ryan Williams teaches a nice course on circuit complexity whose course notes might hopefully be put up online :)

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    $\begingroup$ Eventually, eventually :) $\endgroup$ – Ryan Williams May 12 '14 at 1:46
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In addition to the ones suggested by Arnab, I would also recommend the following book:

Introduction to Circuit Complexity: A Uniform Approach by Heribert Vollmer

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Models of Computation by John E Savage (more info/detail/reviews on amazon). its largely more of an undergraduate textbook but is notable for its circuit-centric point of view. includes many advanced topics/proofs, and its free online, and is very well hyperlinked (TOC, refs, index, etc).

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