Drilling is the riskiest activity in the oil-field exploration and development stage. Real-time measurements are needed to monitor drilling conditions to keep it in the safe operating envelope, guide the drilling system into the most productive zones, and provide information for further stages in the completion of the well. In this article, we describe digital communication systems for drilling, including data transmission and data compression. We begin by describing data transmission techniques used for two systems: mud-pulse telemetry (MPT) and electromagnetic (EM) telemetry. The former conveys data through fluid pressure inside the drilling system, while the latter does so through EM waves that propagate through the formation. These two systems are used in nearly all modern oil-field drilling operations. As the data rates of the two systems are limited to the bits/s regime, data compression is very important, in particular, for the real-time transmission of images. We review recent advances in digital communication-inspired by specific channel and geological models-that have enabled the drilling of record-breaking wells in terms of measured depth and operations in very challenging environments.