Undulator technology is a critical element of modern light sources, including both storage rings and free electron lasers. GeV-level electron beams coupled with modern undulators provide users access to high-quality X-ray beams with tunable wavelengths and in some cases variable polarization.
Here we review the motivation to explore the use of superconducting materials for undulator application, and describe issues and hurdles that must be overcome to bring the technology to maturity. In particular we review superconducting material options, magnetic design issues, field quality and correction schemes, fabrication techniques, and undulator field measurements. Finally, we survey some novel schemes that address specialized applications, including very short-period, narrow gap concepts for single-wavelength applications and variable polarization schemes for soft X-ray facilities.