Digital Manufacturing (DM) refers to the on-going adoption of smarter, more agile manufacturing processes and cyber-physical systems. This includes modern techniques and technologies such as Additive Manufacturing (AM)/3D printing, as well as the Industrial Internet of Things (IIoT) and the broader trend toward Industry 4.0. However, this adoption is not without risks: with a growing complexity and connectivity, so too grows the cyber-physical attack surface. Here, malicious actors might seek to steal sensitive information or sabotage products or production lines, causing financial and reputational loss. Of particular concern are where such malicious attacks may enter the complex supply chains of DM systems as Trojans -- malicious modifications that may trigger their payloads at later times or stages of the product lifecycle. In this work, we thus present a comprehensive overview of the threats posed by Trojans in Digital Manufacturing. We cover both hardware and software Trojans which may exist in products or their production and supply lines. From this, we produce a novel taxonomy for classifying and analyzing these threats, and elaborate on how different side channels (e.g. visual, thermal, acoustic, power, and magnetic) may be used to either enhance the impact of a given Trojan or utilized as part of a defensive strategy. Other defenses are also presented -- including hardware, web-, and software-related. To conclude, we discuss seven different case studies and elaborate how they fit into our taxonomy. Overall, this paper presents a detailed survey of the Trojan landscape for Digital Manufacturing: threats, defenses, and the importance of implementing secure practices.