This paper offers (in two parts) a broad overview of recent developments concerning the use of curved space concepts in amorphous structures. Keeping particularly in mind nonspecialist readers, expository background material is included, wherever appropriate. Part I deals essentially with geometrical modelling, and starts with a brief recapitualtion of the famous model-building exercise due to Bernal. We then discuss the Kleman-Sadoc prescription for realizing amorphous structures as mappings of spherical polytopes (the four-dimensional analogue of spherical polyhedra) onto Euclidean space. Such an approach has not only provided a fast and convenient algorithm, but more importantly, has focused attention on the line defects (disclinations) in amorphous structures. As a result, one is now able to relate these disclinations to the Frank-Kasper lines present in complex alloy structures. In turn, this has led to a qualitative scenario for the transformation of the liquid during a cool-down, into the crystalline or the amorphous state. Part II deals with attempts to provide a quantitative structure to this scenario via gauge theories.