  This thesis aims to serve as an introduction to the theory of quasitilings
for amenable groups. In order to showcase the power of this theory, we focus on
the study of the Sofic L\"uck Approximation Conjecture, which can be proven for
amenable groups by making use of quasitilings. The first four chapters of the
thesis are an exposition of the aforementioned topics, collected from the
literature. After that, in the fifth and final chapter we present some new
results related to the Sofic L\"uck Approximation Conjecture for amenable
groups over discrete valuation rings and number fields.
