  Let $G$ be a locally compact second countable group. We present a
comprehensive treatment of the classical Hopf Decomposition (also known as the
Conservative-Dissipative Decomposition) for general nonsingular $G$-spaces, and
provide several fundamentally different characterizations of it. Subsequently,
we establish a complete structure theorem for totally dissipative nonsingular
$G$-spaces through the construction of \emph{Krengel $G$-spaces}, extending
Krengel's structure theorem on flows.
