An ion of mass-to-charge ratio, m/q, moving under the combined influence of a static magnetic field, B, and an axial quadrupolar electrostatic potential field undergo three "natural" ("normal mode") motions: cyclotron rotation at frequency ω+, magnetron rotation at frequency ω−, and an axial "trapping" oscillation at frequency ωT. Ordinarily, magnetron motion is unobserved, and considered to be undesirable, because ion-neutral collisions result in radial diffusive loss of ions as they roll "downhill" on the magnetron potential surface. However, azimuthal quadrupolar excitation at the ion cyclotron frequency, ωc = qB/m, converts magnetron motion into cyclotron motion, and (for heavy ions) ion-neutral collisions then shrink the cyclotron orbit, leaving a compact ion packet at or near the center of the ICR ion trap. In this paper, we describe briefly the formal basis for such ion axialization, visualization of magnetron motion, the effect of quadrupolar rf excitation amplitude and duration, and how to achieve ion axialization over a wide range of ion m/q values. Applications of the axialization process include ion remeasurement with high (up to 99%) efficiency; prolonged and efficient ion trapping at high collision gas pressure for cooling of electronically, vibrationally, and/or rotationally "hot" ions; enhanced mass resolving power and peak height-to-noise ratio; and enhanced parent ion selectivity (first stage of MS/MS) and enhanced product ion S/N ratio and mass resolving power (second stage of MS/MS). Moreover, because ions may now be axialized, they need not be formed or injected on-axis, thereby making possible continuous injection of externally formed ions, and/or separation between ion and optical axes for photoionization, photodissociation, photodetachment, and optical spectroscopy of trapped ions.