As the performance of digital circuits and the capacity of digital communication channels have advanced steadily over the past decades, digital signals have replaced analog signals in nearly every technological application domain, offering robustness, ever increasing quality, and speed. At the same time, many signals that we perceive audiovisually (sound and video), or measure via instrumentation (e.g., geophysical signals and medical images), are analog in their nature. Consequently, not only are the analog-to-digital (A/D) and digital-to-analog (D/A) conversion processes as crucial as ever, but also the performance of these processes must improve to meet the increasing demands at the digital processing end. Both A/D and D/A conversion is necessarily carried out via analog devices bound by physical limitations such as imprecisions and noise, which present continual technological as well as theoretical challenges to achieving this objective. This article aims to introduce some of the modern mathematical ideas, developments, and challenges in this context from the author's perspective. © 2012 Wiley Periodicals, Inc.