  In this thesis we study the Darboux transformations related to particular Lax
operators of NLS type which are invariant under the action of the so-called
reduction group. Specifically, we study the cases of: 1) the nonlinear
Schr\"odinger equation (with no reduction), 2) the derivative nonlinear
Schr\"odinger equation, where the corresponding Lax operator is invariant under
the action of the $\mathbb{Z}_2$-reduction group and 3) a deformation of the
derivative nonlinear Schr\"odinger equation, associated to a Lax operator
invariant under the action of the dihedral reduction group. These reduction
groups correspond to recent classification results of automorphic Lie algebras.
  We derive Darboux matrices for all the above cases and we use them to
construct novel discrete integrable systems together with their Lax
representations. For these systems of difference equations, we discuss the
initial value problem and, moreover, we consider their integrable reductions.
Furthermore, the derivation of the Darboux matrices gives rise to many
interesting objects, such as B\"acklund transformations for the corresponding
partial differential equations as well as symmetries and conservation laws of
their associated systems of difference equations.
  Moreover, we employ these Darboux matrices to construct six-dimensional
Yang-Baxter maps for all the afore-mentioned cases. These maps can be
restricted to four-dimensional Yang-Baxter maps on invariant leaves, which are
completely integrable; we also consider their vector generalisations.
