  In this thesis we study singular curves on K3 surfaces. Let $\mathcal{B}_g$
denote the stack of polarised K3 surfaces of genus $g$ and set
$p(g,k)=k^2(g-1)+1$. There is a stack $ \mathcal{T}^n_{g,k} \to \mathcal{B}_g$
with fibre over the polarised surface $(X,L)$ parametrising all unramified
morphisms $f: C \to X$, birational onto their image, with $C$ an integral
smooth curve of genus $ p(g,k)-n$ and $f_*C \sim kL$. One can think of $
\mathcal{T}^n_{g,k}$ as parametrising all singular curves on K3 surfaces such
that the normalisation map is unramified (or equivalently such that the curve
has "immersed" singularities).
  The stack $ \mathcal{T}^n_{g,k}$ comes with a natural moduli map $$\eta \; :
\;\mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ to the Deligne-Mumford stack
of curves, defined by forgetting the map to the K3 surface. We first show that
$\eta$ is generically finite (to its image) on at least one component of
$\mathcal{T}^n_{g,k} $, in all but finitely many values of $p(g,k)-n$. We also
consider related questions about the Brill-Noether theory of singular curves on
K3 surfaces as well as the surjectivity of twisted Gaussian maps on
normalisations of singular curves. Lastly, we apply the deformation theory of
$\mathcal{T}^n_{g,k}$ to a seemingly unrelated problem, namely the
Bloch-Beilinson conjectures on the Chow group of points of K3 surfaces with a
symplectic involution.
