According to the World Economic Forum, cyberattacks are considered as one of the most important sources of risk to companies and institutions worldwide. Attacks can target the network, software, and/or hardware. Over the years, much knowledge has been developed to understand and mitigate cyberattacks. However, new threats have appeared in recent years regarding software attacks that exploit hardware vulnerabilities. This article defines these attacks as architectural attacks. Today, both industry and academia have only limited comprehension of architectural attacks, which represents a critical issue for the design of future systems. To this end, this work proposes a new taxonomy, a new attack model, and a complete survey of existing architectural attacks. As a result, it provides the tools to understand architectural attacks in more depth and to start building improved designs and protection mechanisms.