  We prove various iteration theorems for forcing classes related to subproper
and subcomplete forcing, introduced by Jensen. In the first part, we use
revised countable support iterations, and show that 1) the class of subproper,
${}^\omega\omega$-bounding forcing notions, 2) the class of subproper,
$T$-preserving forcing notions (where $T$ is a fixed Souslin tree) and 3) the
class of subproper, $[T]$-preserving forcing notions (where $T$ is an
$\omega_1$-tree) are iterable with revised countable support. In the second
part, we adopt Miyamoto's theory of nice iterations, rather than revised
countable support. We show that this approach allows us to drop a technical
condition in the definitions of subcompleteness and subproperness, still
resulting in forcing classes that are iterable in this way, preserve
$\omega_1$, and, in the case of subcompleteness, don't add reals. Further, we
show that the analogs of the iteration theorems proved in the first part for
RCS iterations hold for nice iterations as well.
