  We establish adiabatic theorems with and without spectral gap condition for
general operators $A(t): D(A(t)) \subset X \to X$ with possibly time-dependent
domains in a Banach space $X$. We first prove adiabatic theorems with uniform
and non-uniform spectral gap condition (including a slightly extended adiabatic
theorem of higher order). In these adiabatic theorems the considered spectral
subsets $\sigma(t)$ have only to be compact -- in particular, they need not
consist of eigenvalues. We then prove an adiabatic theorem without spectral gap
condition for not necessarily (weakly) semisimple eigenvalues: in essence, it
is only required there that the considered spectral subsets $\sigma(t) = \{
\lambda(t) \}$ consist of eigenvalues $\lambda(t) \in \partial \sigma(A(t))$
and that there exist projections $P(t)$ reducing $A(t)$ such that
$A(t)|_{P(t)D(A(t))}-\lambda(t)$ is nilpotent and
$A(t)|_{(1-P(t))D(A(t))}-\lambda(t)$ is injective with dense range in
$(1-P(t))X$ for almost every~$t$. In all these theorems, the regularity
conditions imposed on $t \mapsto A(t)$, $\sigma(t)$, $P(t)$ are fairly mild. We
explore the strength of the presented adiabatic theorems in numerous examples.
And finally, we apply the adiabatic theorems for time-dependent domains to
obtain -- in a very simple way -- adiabatic theorems for operators $A(t)$
defined by symmetric sesquilinear forms.
