Preface.- Introductory Overview.- Part I: Multipactor in a planar gap.- Existence zones for a multipactor discharge.- Introduction.- Distribution of the normal velocity components.- Analysis of the equation of motion.- Stability condition.- Returning electrons.- Energy constraints.- Conclusion.- Generalized phase stability in multipacting.- Introduction.- Distribution of initial velocities and the SEY = 1 boundary.- Stability condition for different points of the multipacting zone.- Other approaches to the phase stability in a flat gap.- Conclusion.- Ping-pong modes.- Introduction.- Boundaries of the ping-pong modes.- Stability boundaries.- Cutoff boundaries.- Lines of equal impact energy.- Boundaries of the two-surface MP and overlapping with the ping-pong MP.- Conclusion.- Simulation of multipactor in a planar gap.- General codes.- Codes ad hoc.- Part II: Multipactor in crossed RF fields.- Effect of the RF cavity magnetic field on multipactor in a gap.- Experimental cavity for 430 MHz.- Inclusion of magnetic field into equations of motion.- Multipactor near the cavity equator.- Introduction.- Fields near equator.- Dependence of the upper arc fields on the lower arc geometry.- Equations of motion.- Condition of stability.- Multipacting maps.- Deviations from the elliptic geometry.- Comparison with experiment.- Conclusion.- Belomestnykh.- One-point multipactor in crossed fields of RF cavities.- Introduction.- Fields and equations of motion in a known geometry with one-point MP.- Comparison of analytical calculations with simulations and experiment.- Influence of change of the surface electric field. Multipactor map.- Phase and space stability. Traveling multipactor.- Comparison of the equations of motion for MP1 and MP2.- Discussion and conclusions.- Part III: Multipacting-free cavities and transitions between cavities and beampipes.- Optimized shape cavities free of MP.- Introduction.- Elliptic geometry and surface fields.- Some definitions.- Method of optimization.- More constraints to the shape of the elliptic cavity.- An example of optimization for the TESLA cavity.- An example of optimization for the SNS elliptic cavity with bgeo = 0.81.- Multipactor consideration.- Conclusion.- Multipacting-free transitions between cavities and beam-pipes. Theorem on minimal electric field.- Introduction.- Cavity with transition from iris to a larger diameter beam-pipe.- Cavity with a tapered end port.- Mechanism of the motion.- Conclusion.