  A charged particle in a magnetic field possesses discrete energy levels
associated with particle's rotation around the field lines. The radiative
transitions between these levels are the well-known cyclotron transitions. We
show that a bound complex of particles with a non-zero net charge displays
analogous transitions between the states of confined motion of the entire
complex in the field. The latter {\em bound-ion cyclotron transitions} are
affected by a coupling between the collective and internal motions of the
complex and, as a result, differ from the transitions of a "reference" bare ion
with the same mass and charge. We analyze the cyclotron transitions for complex
ions by including the coupling within a rigorous quantum approach. Particular
attention is paid to comparison of the transition energies and oscillator
strengths to those of the bare ion. Selection rules based on integrals of
collective motion are derived for the bound-ion cyclotron transitions
analytically, and the perturbation and coupled-channel approaches are developed
to study the transitions quantitatively. Representative examples are considered
and discussed for positive and negative atomic and cluster ions.
