Since Rijndael was chosen as the Advanced Encryption Standard, improving upon 7-round attacks on the 128-bit key variant or upon 8-round attacks on the 192/256-bit key variants has been one of the most difficult challenges in the cryptanalysis of block ciphers for more than a decade. In this paper we present a novel technique of block cipher cryptanalysis with bicliques, which leads to the following results: – The first key recovery attack on the full AES-128 with computational complexity 2126.1 . – The first key recovery attack on the full AES-192 with computational complexity 2189.7 . – The first key recovery attack on the full AES-256 with computational complexity 2254.4 . – Attacks with lower complexity on the reduced-round versions of AES not considered before, including an attack on 8-round AES-128 with complexity 2124.9 . – Preimage attacks on compression functions based on the full AES versions. In contrast to most shortcut attacks on AES variants, we do not need t...