Building on an earlier work, we show how the conformal group can be applied in both R ⁿ and C ⁿ to shed new light on results in Clifford analysis. This includes setting up a Bergman kernel over unbounded domains in C ⁿ and exhibiting a conformal covariance for harmonic measure. Many of the results arise from the conformal covariance of cells of harmonicity and special real n-dimensional manifolds lying in C ⁿ . These manifolds form natural generalizations of domains in R ⁿ . We also use conformal covariance to adapt existing results to illustrate the L ²-boundedness of the double-layer potential operator over Lipschitz perturbations of the sphere, to describe a conformal covariance associated to mutually commuting operators over a real Banach algebra, and to set up boundary value problems for a particular inhomogeneous equation with Laplacian as the principal part.