In the last decade one observes an increasing interest in the study of more refined structures who model more accurately numerous physical problems. While the classic Dirac operator can be used to describe structures with $Z_2$-grading it is not adequate for other situations (for example, problems in optics (polarisation filters) which require analysis of $SU(3)$-symmetries, or for Boson Fock spaces in quantum superspaces). For all of these cases, the classic Dirac operator and related function theories do not provide a satisfactory representation. In this talk we are particularly interested in $SU(3)$-symmetries. This problem is nowadays mainly modelled by means of supersymmetry. However, we use another approach, based on ternary generalised Clifford algebras, or $Z_3$-graded cubic algebras. We provide a complete algebraic description of such algebras and develop the necessary analytic tools for a consistent function theory.