  Amorphous solids show surprisingly universal behaviour at low temperatures.
The prevailing wisdom is that this can be explained by the existence of
two-state defects within the material. The so-called standard tunneling model
has become the established framework to explain these results, yet it still
leaves the central question essentially unanswered -- what are these two-level
defects? This question has recently taken on a new urgency with the rise of
superconducting circuits in quantum computing, circuit quantum electrodynamics,
magnetometry, electrometry and metrology. Superconducting circuits made from
aluminium or niobium are fundamentally limited by losses due to two-level
defects within the amorphous oxide layers encasing them. On the other hand,
these circuits also provide a novel and effective method for studying the very
defects which limit their operation. We can now go beyond ensemble measurements
and probe individual defects -- observing the quantum nature of their dynamics
and studying their formation, their behaviour as a function of applied field,
strain, temperature and other properties. This article reviews the plethora of
recent experimental results in this area and discusses the various theoretical
models which have been used to describe the observations. In doing so, it
summarises the current approaches to solving this fundamentally important
problem in solid-state physics.
