  Quandle representations are homomorphisms from a quandle to the group of
invertible matrices on some vector space taken with the conjugation operation.
We study certain families of quandle representations. More specifically, we
introduce the notion of regular representation for quandles, investigating in
detail the regular representations of dihedral quandles and \emph{completely
classifying} them. Then, we study representations of cyclic quandles, giving
some necessary conditions for irreducibility and providing a complete
classification under some restrictions. Moreover, we provide various
counterexamples to constructions that hold for group representations, and show
to what extent such theory has the same properties of the representation theory
of finite groups. In particular, we show that Maschke's theorem does not hold
for quandle representations.
