Given a row-finite $k$-graph $��$ with no sources we investigate the $K$-theory of the higher rank graph $C^*$-algebra, $C^*(��)$. When $k=2$ we are able to give explicit formulae to calculate the $K$-groups of $C^*(��)$. The $K$-groups of $C^*(��)$ 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 $K_0(C^*(��))$ and $K_1(C^*��))$ are equal when $C^*(��)$ is unital, and for $k=2$ we determine the position of the class of the unit of $C^*(��)$ in $K_0(C^*(��))$.

23 pages. To appear in the New York Journal of Mathematics (http://nyjm.albany.edu:8000/). Revisions include: a different numbering system for sections, theorems and related parts; correction of typographical errors; re-organisation of results; and addition of examples (Section 5)