  Given a row-finite $k$-graph $\Lambda$ with no sources we investigate the
$K$-theory of the higher rank graph $C^*$-algebra, $C^*(\Lambda)$. When $k=2$
we are able to give explicit formulae to calculate the $K$-groups of
$C^*(\Lambda)$. The $K$-groups of $C^*(\Lambda)$ for $k>2$ can be calculated
under certain circumstances and we consider the case $k=3$. We prove that for
arbitrary $k$, the torsion-free rank of $K_0(C^*(\Lambda))$ and
$K_1(C^*\Lambda))$ are equal when $C^*(\Lambda)$ is unital, and for $k=2$ we
determine the position of the class of the unit of $C^*(\Lambda)$ in
$K_0(C^*(\Lambda))$.
