  In this paper we establish a criterion for the triviality of the
$C^1$-centralizer for vector fields and flows. In particular we deduce the
triviality of the centralizer at homoclinic classes of $C^r$ vector fields
($r\ge 1$). Furthermore, we show that the set of flows whose $C^1$-centralizer
is trivial include: (i) $C^1$-generic volume preserving flows, (ii)
$C^2$-generic Hamiltonian flows on a generic and full Lebesgue measure set of
energy levels, and (iii) $C^1$-open set of non-hyperbolic vector fields (that
admit a Lorenz attractor). We also provide a criterion for the triviality of
the $C^0$-centralizer of continuous flows.
