The Kondo effect assumes a central role in condensed matter physics. It describes the exchange scattering of electrons with a localized moment and occurs in a wide range of settings from rare earth-based intermetallics to nanotubes and semiconductor heterostructures. A conceptually simple extension of the model, known as the two-channel Kondo model, can give rise to singular scattering and the formation of an unconventional metal. Although the observation of the standard Kondo effect has become ubiquitous, its two-channel counterpart has been proven difficult to realize. This article reviews attempts, challenges, and successes in realizing the two-channel Kondo effect in artificial structures and real quantum materials.