Processor design and manufacturing is often done globally, involving multiple companies, some of which can be untrustworthy. This lack of trust leads to the threat of malicious modifications like Hardware Trojans. Hardware Trojans can cause drastic consequences and even endanger human lives. Hence, effective countermeasures against Hardware Trojans are urgently needed. To develop countermeasures, Hardware Trojans and their properties have to be understood well. For this reason, we describe and characterize Hardware Trojans in detail in this paper. We perform a theoretical analysis of Hardware Trojans for processors. Afterwards, we present a new classification of processor constituents, which can be used to derive several triggers and payloads and compare them with previously published Hardware Trojans. This shows in detail possible attack vectors for processors and gaps in existing processor Hardware Trojan landscape. No previous work presents such a detailed investigation of Hardware Trojans for processors. With this work, we intend to improve understanding of Hardware Trojans in processors, supporting the development of new countermeasures and prevention techniques.