Zeeman deceleration is an experimental technique which allows for the manipulation of open-shell atoms and molecules in a supersonic beam thus producing mK-cold, velocity-tunable beams of particles in selected quantum states. The method relies on the Zeeman interaction between paramagnetic particles and time-varying, inhomogeneous magnetic fields generated by pulsing high currents through an array of solenoid coils. This thesis describes the construction and implementation of a supersonic beam setup including a 12-stage Zeeman decelerator. The Zeeman decelerator follows an original design that makes it possible to replace individual deceleration coils. Using ground-state hydrogen atoms as a test system, it is shown that the transverse acceptance in a Zeeman decelerator can be significantly increased by generating a rather low, temporally varying quadrupole field in one of the solenoid coils. An electron-impact source was constructed and optimised enabling, for the first time, the Zeeman deceleration of metastable helium atoms in the 23S1 state, with an up to 40 % decrease in the kinetic energy of the beam. It is shown that the pulse duration for electron-impact excitation needs to be matched to the acceptance of the decelerator in order to attain a good contrast between the decelerated and undecelerated parts of the beam. Experimental results are rigorously analysed and interpreted using three-dimensional numerical particle trajectory simulations. A phase-space model provides, for the first time, a means to estimate the six-dimensional phase-space acceptance in a Zeeman decelerator and to find optimum parameter sets for improved Zeeman deceleration schemes. The approach also reveals a hitherto unconsidered velocity dependence of the phase stability which is ascribed mainly to the rise and fall times of the current pulses that generate the magnetic fields inside the deceleration coils. In the future, it is planned to combine the Zeeman decelerator with a source of cold atomic and molecular ions to study chemical collisions at low temperatures. A hybrid magnetic guide consisting of permanent magnet assemblies (Halbach arrays) in hexapole configuration and a set of current-carrying wires is proposed and simulated as an interface between these setups. The design promises very efficient velocity selection, a high degree of quantum-state selection and a nearly complete removal of residual carrier gas. Prospects for using magnetic hexapole focusing in front of the Zeeman decelerator are discussed. The work represents a major step towards the study and control of chemical reactivity of paramagnetic species in the low-temperature regime and it will help in the testing of fundamental chemical reaction theories.