The discovery of magnetic and compositional effects in the low temperature properties of multicomponent glasses has prompted the need to extend the standard two-level systems (2LSs) tunneling model. A possible extension [1] assumes that a subset of tunneling quasi-particles is moving in a threewelled potential (TWP) associated with the ubiquitous inhomogeneities of the disordered atomic structure of the glass. We show that within an alternative, cellular description of the intermediate-range atomic structure of glasses the tunneling TWP can be fully justified. We then review how the experimentally discovered magnetic effects can be explained within the approach where only localized atomistic tunneling 2LSs and quasi-particles tunneling in TWPs are allowed. We discuss the origin of the magnetic effects in the heat capacity, dielectric constant (real and imaginaryi parts), polarization echo and SQUID magnetization in several glassy systems. We conclude by commenting on a strategy to reveal the mentioned tunneling states (2LSs and TWPs) by means of atomistic computer simulations and discuss the microscopic nature of the tunneling states in the context of the potential energy landscape of glass-forming systems.arXiv:1508.02424v1 [cond-mat.dis-nn] 10 Aug 2015 1 IntroductionThe physics of glasses, especially at low temperatures, continues to attract considerable interest. At low temperatures glasses are believed to be characterized by low-energy excitations going under the name of tunneling systems, or states (TSs) which are normally described in terms of double-welled potentials (DWPs) and two-level systems (2LSs) with energy asymmetry and tunneling barrier uniformly distributed in the amorphous solid [2,3]. Little is still known about the nature of the TSs, but the general consensus is still that the intermediate-range atomic structure of glasses is well described by Zachariasen's 1932 continuous random network model [4,5] (thus, homogeneously disordered like for a liquid) and the 2LSs arise out of two slightly similar, localized atomic configurations. With this characterization the 2LSs have been employed in the 1970s and 80s to explain with some success the anomalies in the properties of glasses at low temperatures. Recently, the field has been witnessing a renaissance since the discovery that there are unexpected magnetic effects in insulating glasses [6,7,8,9] and that the TSs are responsible for the decoherence of phase qubits made of superconducting Josephson-junctions [10,11,12,13,14]. The 2LSs are thought to be ubiquitous in the junction's tunneling barrier which, though very thin, is considered to be amorphous and described by Zachariasen-Warren's model [4,5]. On the other hand, the low-temperature glasses have also been studied as a paradigm where to conduct research on the physics of aging so common to many disordered systems out of equilibrium [15,16]. In this context, the relevant degrees of freedom have also been described in terms of 2LSs, with some success [17]. Yet, deviations from the behaviour pred...