  One may think of a $d$-class association scheme as a $(d+1)$-dimensional
matrix algebra over $\mathbb{R}$ closed under Schur products. In this context,
an imprimitive scheme is one which admits a subalgebra of block matrices, also
closed under the Schur product. Such systems of imprimitivity provide us with
quotient schemes, smaller association schemes which are often easier to
understand, providing useful information about the structure of the larger
scheme. For any association scheme we find a basis of $d+1$ idempotent matrices
for the algebra. A cometric scheme is one whose idempotent basis may be ordered
$E_0,E_1,...,E_d$ with polynomials $f_0,f_1,...,f_d$ giving $f_i\circ(E_1)=E_i$
and deg$(f_i)=i$ for each $i$. Throughout this thesis we are primarily
interested in three goals: building new examples of cometric schemes, drawing
connections between cometric schemes and other objects, and finding new
realizability conditions on feasible parameter sets --- using these conditions
to rule out open parameter sets when possible. After introducing association
schemes, this thesis focuses on a few recent results regarding cometric schemes
with small $d$. We begin by examining the matrix algebra, looking for low rank
positive semidefinite matrices with few distinct entries and later establishing
new conditions on realizable parameter sets. We then focus on imprimitive
examples of both 3- and 4-class cometric schemes, generating new examples of
the former while building realizability conditions for both. In each case, we
examine the related $t$-distance sets, giving conditions which work towards
equivalence; in the case of 3-class $Q$-antipodal schemes an equivalence is
established. We conclude by partially extending a result of Brouwer and Koolen
concerning the connectivity of graphs arising from metric schemes.
