In this work, we present a series of structure transformation attacks on LLM alignment, where we encode natural language intent using diverse syntax spaces, ranging from simple structure formats and basic query languages (e.g., SQL) to new novel spaces and syntaxes created entirely by LLMs. Our extensive evaluation shows that our simplest attacks can achieve close to a 90% success rate, even on strict LLMs (such as Claude 3.5 Sonnet) using SOTA alignment mechanisms. We improve the attack performance further by using an adaptive scheme that combines structure transformations along with existing content transformations, resulting in over 96% ASR with 0% refusals. To generalize our attacks, we explore numerous structure formats, including syntaxes purely generated by LLMs. Our results indicate that such novel syntaxes are easy to generate and result in a high ASR, suggesting that defending against our attacks is not a straightforward process. Finally, we develop a benchmark and evaluate existing safety-alignment defenses against it, showing that most of them fail with 100% ASR. Our results show that existing safety alignment mostly relies on token-level patterns without recognizing harmful concepts, highlighting and motivating the need for serious research efforts in this direction. As a case study, we demonstrate how attackers can use our attack to easily generate a sample malware and a corpus of fraudulent SMS messages, which perform well in bypassing detection.