  We review some of the recent developments and prove new results concerning
frames and Bessel systems generated by iterations of the form $\{A^ng: g\in
G,\, n=0,1,2,\dots \}$, where $A$ is a bounded linear operators on a separable
complex Hilbert space $ \mathcal{H} $ and $G$ is a countable set of vectors in
$ \mathcal{H} $. The system of iterations mentioned above was motivated from
the so called dynamical sampling problem. In dynamical sampling, an unknown
function $f$ and its future states $A^nf$ are coarsely sampled at each time
level $n$, $0\leq n< L$, where $A$ is an evolution operator that drives the
system. The goal is to recover $f$ from these space-time samples.
