In this thesis, we study the representation theory of Lie colour algebras. Our strategy follows the work of G. Benkart, C. L. Shader and A. Ram in 1998, which is to use the Brauer algebras which appear as the commutant of the orthosymplectic Lie colour algebra when they act on a k-fold tensor product of the standard representation. We give a general combinatorial construction of highest weight vectors using tableaux, and compute characters of the irreducible summands in some borderline cases. Along the way, we prove the RSK-correspondence for tableaux and the PBW theorem for Lie colour algebras.