  This paper is concerned with the existence of conformal metrics of the disk
with prescribed Gaussian and geodesic curvatures. Being more specific, given
nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h:
\partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a
conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures,
respectively. This is the natural analogue of the classical Nirenberg problem
posed on the disk. As we shall see, both curvatures play a role in the
existence of solutions. Indeed we are able to give existence results under
conditions that involve $K$ and $H$, where $H$ denotes the harmonic extension
of $h$. The proof is based on the computation of the Leray-Schauder degree in a
compact setting.
