  We study certain dynamical systems which leave invariant an indefinite
quadratic form via semigroups or evolution families of complex symmetric
Hilbert space operators. In the setting of bounded operators we show that a
$\mathcal{C}$-selfadjoint operator generates a contraction $C_0$-semigroup if
and only if it is dissipative. In addition, we examine the abstract Cauchy
problem for nonautonomous linear differential equations possessing a complex
symmetry. In the unbounded operator framework we isolate the class of
\emph{complex symmetric, unbounded semigroups} and investigate Stone-type
theorems adapted to them. On Fock space realization, we characterize all
$\mathcal{C}$-selfadjoint, unbounded weighted composition semigroups. As a
byproduct we prove that the generator of a $\mathcal{C}$-selfadjoint, unbounded
semigroup is not necessarily $\mathcal{C}$-selfadjoint.
