Planar quadrupole ion traps have been analyzed through numerical and analytic solutions of Laplaces equation. These involve either one or more conducting rings or their analogs, a hole in one or more conducting sheets. The leading terms in the potential are harmonic, corresponding to the Paul trap, but with coefficients that reduce their efficiency and for some traps, the anharmonic terms can be suppressed to eighth-order. Stable ion trapping is predicted for all electrode configurations possessing radial and axial symmetry. A three-hole microtrap with an inner hole radius of 80 m trapped from one to many (dense clouds) laser-cooled Ba+ ions where the two-ion distance is compressed to 1 m, allowing new experiments in quantum optics. Also, arrays of traps for optical clocks are contemplated using photolithographic fabrication. © 1992 The American Physical Society.