This thesis presents numerical optimization of geometries of axially symmetric ion trap mass analyzers. The motivation for this thesis is two fold. First is to demonstrate how the automated scheme can be applied to achieve geometry parameters of axially symmetric ion traps for a desired field configuration. Second is, through the Geometries investigated in this thesis, to present practically achievable geometries for mass spectroscopists to use. Here the underlying thought has been to keep the design simple for ease of fabrication (with the possibility of miniaturization) and still ensure that the performance of these analyzers is similar to the stretched geometry Paul traps. Five geometries have been taken up for investigation: one is the well known Cylindrical ion trap (CIT), three are new geometries and the last is the Paul trap under development in our laboratory. Two of these newer geometries have a step in the region of the midline of the cylindrical ring electrode (SRIT) and the third geometry has a step in its endcap electrodes (SEIT). The optimization has been carried out around deferent objective functions composed of the desired weights of higher order multiples. The Nelder-Mead simplex method has been used to optimize trap geometries. The multipoles included in the computations are quadrupole, octopole, dodecapole, hexadecapole,ikosipole and tetraikosipole having weights A2, A4, A6, A8, A10 and A12, respectively.Poincare sections have been used to understand dynamics of ions in the traps investigated. For the CIT, it has been shown that by changing the aspect ratio of the trap the harmful ejects of negative dodecapole superposition can be eliminated, although this results in a large positive A4=A2 ratio. Improved performance of the optimized CIT is suggested by the ion dynamics as seen in Poincare sections close to the stability boundary. With respect to the SRIT, two variants have been investigated. In the first geometry, A4=A2 and A6=A2 have been optimized and in the second A4=A2, A6=A2 and A8=A2 have been optimized; in both cases, these ratios have been kept close to their values reported for stretched hyperboloid geometry Paul traps. In doing this, however, it was seen that the weights of still higher order multipole not included in the objective function, A10=A2 and A12=A2, are high; additionally, A10=A2 has a negative sign. In spite of this, for both these configurations, the Poincare sections predict good performance. In the case of the SEIT, a geometry was obtained for which A4=A2 and A6=A2 are close to their values in the stretched geometry Paul trap and the higher even multipole (A8=A2, A10=A2 and A12=A2) are all positive and small in magnitude. The Poincare sections predict good performance for this con¯guration too. Direct numerical simulations of coupled nonlinear axial/radial dynamics also predict good performance for the SEIT, which seems to be the most promising among the geometries proposed here. Finally, for the Paul trap under development in our laboratory, Poincare sections and numerical simulations of coupled ion dynamics suggest a stretch of 79:7% is the best choice.