  Inspired by Katok's examples of Finsler metrics with a small number of closed
geodesics, we present two results on Reeb flows with finitely many periodic
orbits. The first result is concerned with a contact-geometric description of
magnetic flows on the 2-sphere found recently by Benedetti. We give a simple
interpretation of that work in terms of a quaternionic symmetry. In the second
part, we use Hamiltonian circle actions on symplectic manifolds to produce
compact, connected contact manifolds in dimension at least five with
arbitrarily large numbers of periodic Reeb orbits. This contrasts sharply with
recent work by Cristofaro-Gardiner, Hutchings and Pomerleano on Reeb flows in
dimension three. With the help of Hamiltonian plugs and a surgery construction
due to Laudenbach we reprove a result of Cieliebak: one can produce Hamiltonian
flows in dimension at least five with any number of periodic orbits; in
dimension three, with any number greater than one.
