  We determine simplicity criteria in characteristics 0 and $p$ for a
ubiquitous class of iterated skew polynomial rings in two indeterminates over a
base ring. One obstruction to simplicity is the possible existence of a
canonical normal element $z$. In the case where this element exists we give
simplicity criteria for the rings obtained by inverting $z$ and the rings
obtained by factoring out the ideal generated by $z$. The results are
illustrated by numerous examples including higher quantized Weyl algebras and
generalizations of some low-dimensional symplectic reflection algebras.
