The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in are of particular interest. In this paper we discuss various examples. In the case of and we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.