The theory of error-correcting codes, originally developed as a fundamental technique for a systematic study of communications systems, has served as a pivotal tool in major areas of mathematics, computer science and electrical engineering. Understanding problems through a "coding lens" has consistently led to breakthroughs in a wide spectrum of research areas, often seemingly foreign from coding theory, including discrete mathematics, geometry, cryptography, signal processing, algorithms and complexity, to name a few. This talk will focus on the role of coding theory in explicit construction problems in sparse recovery and signal processing.