When are vector fields Hamiltonian?

Dynamical systems can be quantised only if they are Hamiltonian. This prompts the question from which our talk gets its title. We show how the simple predator-prey equation and the damped harmonic oscillator can be considered to be Hamiltonian with respect to an infinite number of non-standard Poisson brackets. This raises some interesting questions about the nature of quantisation. Questions which are valid even for flows which possess a canonical structure.