This paper analyses the concept of Self-Sovereign Identity (SSI), an emerging approach for establishing digital identity, in the context of the Internet of Things (IoT). We contrast existing approaches for identity on the Internet, such as cloud-based accounts and digital certificates, with SSI standards such as Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs). To the best of our knowledge, this is the first thorough comparison of these approaches. The benefits and challenges of using DIDs and VCs to identify and authenticate IoT devices and their respective users are discussed. In the end, we establish that SSI, with its owner-centric, privacy-aware and decentrailized approach, provides a viable and attractive option for secure identification of IoT devices and users.