The Internet of Things (IoT) represents a growing aspect of how entities, including humans and organizations, are likely to connect with others in their public and private interactions. The exponential rise in the number of IoT devices, resulting from ever-growing IoT applications, also gives rise to new opportunities for exploiting potential security vulnerabilities. In contrast to conventional cryptosystems, frameworks that incorporate fine-grained access control offer better opportunities for protecting valuable assets, especially when the connectivity level is dense. Functional encryption is an exciting new paradigm of public-key encryption that supports fine-grained access control, generalizing a range of existing fine-grained access control mechanisms. This survey reviews the recent applications of functional encryption and the major cryptographic primitives that it covers, identifying areas where the adoption of these primitives has had the greatest impact. We first provide an overview of different application areas where these access control schemes have been applied. Then, an in-depth survey of how the schemes are used in a multitude of applications related to IoT is given, rendering a potential vision of security and integrity that this growing field promises. Towards the end, we identify some research trends and state the open challenges that current developments face for a secure IoT realization.