Photodetectors play key roles in many applications such as remote sensing, night vision, reconnaissance, medical imaging, thermal imaging, and chemical detection. Several properties such as performance, reliability, ease of integration, cost, weight, and form factor are all important in determining the attributes of photodetectors for particular applications. While a number of materials have been used over the past several decades to address photodetection needs across the electromagnetic spectrum, the advent of nanomaterials opens new possibilities for photodetectors. In particular, carbon-based nanomaterials such as carbon nanotubes (CNTs) and graphene possess unique properties that have recently been explored for photodetectors. Here, we review the status of the field, presenting a broad coverage of the different types of photodetectors that have been realized with CNTs, placing particular emphasis on the types of mechanisms that govern their operation. We present a comparative summary of the main performance metrics for such detectors, and an outlook for performance improvements.2