Over the last decades it has become routine to form beams of positrons and antiprotons to produce trapped samples of both species for a variety of purposes. Positrons can captured efficiently and in such quantities to form dense, single component plasmas useful for antihydrogen formation. The latter was made possible by developing techniques for dynamically capturing and then cooling antiprotons ejected from the Antiproton Decelerator at CERN. This talk will review recent advances which have included the first demonstration of the trapping of antihydrogen. Trapping, which is thought to be a pre-requisite for spectroscopic comparisons of antihydrogen with hydrogen, causes extra complications by introducing a magnetic minimum neutral atom trap in the same region that the antiparticles are held. We will describe how clouds of antiprotons can be compressed and evaporatively cooled in preparation for antihydrogen formation in such an environment, and how this allows a small fraction of the anti-atoms we create to be held in the trap. We will discuss the physics motivations for undertaking such difficult experiments.