At very low temperatures (below 1-10 K), large family of amorphous solids have various similar low-temperature properties.
Although there are various theoretical models, neither glass transition nor low-temperature properties of glassy solids are well understood on the fundamental physics level.
Amorphous solids is an important area of condensed matter physics aiming to understand these substances at high temperatures of glass transition and at low temperatures towards absolute zero. From 1970s, low-temperature properties of amorphous solids were studied experimentally in great detail.
 For all of these substances, specific heat has a (nearly) linear dependence as a function of temperature, and thermal conductivity has nearly quadratic temperature dependence. These properties are conventionally called anomalous being very different from properties of crystalline solids.
On the phenomenological level, many of these properties were described by a collection of tunneling two-level systems.
Nevertheless, the microscopic theory of these properties is still missing after more than 50 years of the research.
Remarkably, a dimensionless quantity of internal friction is nearly universal in these materials. 
 This quantity is a dimensionless ratio (up to a numerical constant) of the phonon wavelength to the phonon mean free path. Since the theory of tunneling two-level states (TLSs) does not address the origin of the density of TLSs, this theory cannot explain the universality of internal friction, which in turn is proportional to the density of scattering TLSs. The theoretical significance of this important and unsolved problem was highlighted  by Anthony Leggett.