The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2den">de"> of families de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">Sden">de"> of bounded functions on de9fbd" title="Click to view the MathML source">Dden">de">. Symbol classes de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">Sden">de"> that generate important classical commutative and non-commutative Toeplitz algebras in den">de"> are of particular interest. In this paper we discuss various examples. In the case of den">de"> and den">de"> we characterize all irreducible representations of the resulting Toeplitz operator C⁎den">de">-algebras. Their Calkin algebras are described and index formulas are provided.