  This thesis gives a complete description of the Grothendieck group and
divisor class group for large families of two and three dimensional
singularities. The main results presented throughout, and summarised in Theorem
8.1.1, give an explicit description of the Grothendieck group and class group
of Kleinian singularities, their deformations, and compound Du Val (cDV)
singularities in a variety of settings. For such rings R, the main results
assert that there exists an isomorphism between $G_0(R)$ and $\mathbb{Z} \oplus
\mathrm{Cl}(R)$, and the class group is explicitly presented.
  More precisely, we establish these results for 2-dimensional deformations of
global type A Kleinian singularities, 3-dimensional isolated complete local cDV
singularities admitting a noncommutative crepant resolution, any 3-dimensional
type A complete local cDV singularity, polyhedral quotient singularities (which
are non-isolated), and any isolated cDV singularity admitting a minimal model
with only type cAn cDV singularities. We also study various complex reflection
groups in the setting of symplectic quotient singularities, for which this
isomorphism does not hold, and conjecture based on computer evidence that the
reduced Grothendieck group in the case of the symmetric group has size $n!$.
