The physics of glassy systems at low temperatures differs in striking and unexpected ways from that of their crystalline counterparts, in that thermal, transport and dynamic response properties exhibit unusual temperature variation. Atomic or molecular tunneling centers are believed to be at the origin of these phenomena, which show a remarkable degree of universality across a wide spectrum of diverse amorphous substances, ranging from window glass to polystyrene. We describe recent advances in understanding these phenomena, using a microscopic modelling approach. It has for the first time allowed to identify a mechanism responsible for the occurrence of tunneling systems, to characterize them quantitatively, and to provide a simple and transparent explanation for the unusual universality of glassy low temperature anomalies.