A modular multistage Zeeman decelerator consisting of an array of solenoids has been developed to control the velocity of cold, paramagnetic atoms and molecules in supersonic beams. The translational motion these atoms and molecules was manipulated with high precision by exploiting the interaction between their magnetic moments and timedependent inhomogeneous magnetic elds. Deceleration was achieved by pulsing high currents (up to 300 A) through the solenoids of the decelerator with pulse sequences calculated to preserve the phase-space density of the atomic and molecular samples while removing a preset amount of kinetic energy. For the operation of the multistage Zeeman decelerator, solenoids and electronic units were developed to generate strong magneticeld pulses (> 2 T) with rise and fall times of only a few microseconds. An accurate three-dimensional particle-trajectory simulation program has been developed to analyze the deceleration process and optimize its e ciency. This software was thoroughly tested by systematic comparison with experimental results and enabled, in combination with measurements, the complete characterization of the spatial and velocity distributions of the decelerated and trapped atoms for di erent sets of experimental conditions. The program was essential to achieve a complete interpretation of the dynamics of the deceleration process and to devise a strategy to e ciently load the decelerated atoms and molecules into magnetic quadrupole traps. In a rst experiment, hydrogen atoms seeded in a supersonic expansion of Kr have been decelerated from an initial velocity of 435 m/s to nal velocities as low as 107 m/s in a 12-stage Zeeman decelerator. The operation of the decelerator has been analyzed by comparing the results of numerical particle-trajectory simulations with those of independent measurements of the velocity distribution by ion time-ofight mass spectrometry following photoionization of the decelerated atoms. The velocity distribution of the decelerated atom cloud was found to have a half width at half maximum of 25± 12 m/s corresponding to a temperature of ∼ 40 mK. The phase stability of the multistage-Zeeman-deceleration process has been investigated by numerical particle-trajectory simulations and experimental measurements on deuterium atoms in a 24-stage Zeeman decelerator. A one-dimensional analytical model of the phase stability in a multistage Zeeman decelerator was adapted from a one-dimensional model of phase stability in a multistage Stark decelerator and compared with the results of oneand three-dimensional particle-trajectory simulations. The comparison, which included the analysis of the e ects of nite switch-on and switch-o times of the deceleration pulses, revealed that transverse e ects in the decelerator lead to a considerable reduction of the