The creation of cold antihydrogen atoms by the controlled combination of positrons and antiprotons has opened up a new window on fundamental physics. More recently, techniques have been developed that allow some antihydrogen atoms to be created at low enough kinetic energies that they can be held inside magnetic minimum neutral atom traps. With confinement times of many minutes possible, it has become feasible to perform experiments to probe the properties of the antiatom for the first time. We review the experimental progress in this area, outline some of the motivation for studying basic aspects of antimatter physics and provide an outlook of where we might expect this field to go in the coming years.