Edge computing aims to address the challenges associated with communicating and transferring large amounts of data generated remotely to a data center in a timely and efficient manner. A central pillar of edge computing is local (i.e., at- or near-source) data processing capability so that data transfer to a data center for processing can be minimized. Data compression at the edge is therefore a natural component of edge workflows. We present a survey of data compression algorithms with a focus on edge computing. Not all compression algorithms can accommodate the data type heterogeneity, tight processing and communication time constraints, or energy efficiency requirement characteristics of edge computing. We discuss specific examples of compression algorithms that are being explored in the context of edge computing. We end our review with a brief survey of emerging quantum compression techniques that are of importance in quantum information processing, including the proposed concept of quantum edge computing.