Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C*-algebra, C*(Λ). When k=2 we are able to give explicit formulae to calculate the K-groups of C*(Λ). The K-groups of C*(Λ) 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 K₀(C*(Λ)) and K₁(C*(Λ)) are equal when C*(Λ) is unital, and for k=2 we determine the position of the class of the unit of C*(Λ) in K₀(C*(Λ)).

Research supported by the European Union Research Training Network in Quantum Spaces -- Noncommutative Geometry.