Superconducting undulators (SCUs) with a period >15 mm can offer a much higher on-axis undulator field B₀ than state-of-the-art cryogenic permanent magnet undulators with the same period and vacuum gap. The commissioned NbTi planar SCUs for user operation in the Karlsruhe Institute of Technology synchrotron and the advanced photon source storage ring are operated stably without quenches, producing outperformed photon flux in the high energy part of the hard x-ray spectrum. Another potential advantage of deploying SCU is its radiation hardness, a crucial characteristic for being used in free electron lasers (FELs) driven by high repetition rate superconducting linear accelerators (LINACs) and diffraction limited storage rings (DLSRs) with small vacuum gap and large averaged beam current. The development of shorter period but high field SCU is an important mission in an EU founded CompactLight project as this technology would reduce both the length of undulators and the length of LINACs. This review paper first overviews the research and development of SCUs worldwide from the late 1970s to 2021, then presents the SCU design requirements and compares the theory limits of different types of planar and helical SCUs, and finally reviews the technical challenges including the SCU cryostat, the magnetic field measurement, the integral/local field correction and the high-temperature superconductor (HTS) challenges and prospects the research needs for SCUs.