Progress in the ability to accumulate and cool positrons and antiprotons is enabling new scientific and technological opportunities. The driver for this work is plasma physics research—developing new ways to create and manipulate antimatter plasmas. An overview is presented of recent results and near-term goals and challenges. In atomic physics, new experiments on the resonant capture of positrons by molecules provide the first direct evidence that positrons bind to "ordinary" matter (i.e., atoms and molecules). The formation of low-energy antihydrogen was observed recently by injecting low-energy antiprotons into a cold positron plasma. This opens up a range of new scientific opportunities, including precision tests of fundamental symmetries such as invariance under charge conjugation, parity, and time reversal, and study of the chemistry of matter and antimatter. The first laboratory study of electron-positron plasmas has been conducted by passing an electron beam through a positron plasma. The next major step in these studies will be the simultaneous confinement of electron and positron plasmas. Although very challenging, such experiments would permit studies of the nonlinear behavior predicted for this unique and interesting plasma system. The use of trap-based positron beams to study transport in fusion plasmas and to characterize materials is reviewed. More challenging experiments are described, such as the creation of a Bose-condensed gas of positronium atoms. Finally, the future of positron trapping and beam formation is discussed, including the development of a novel multicell trap to increase by orders of magnitude the number of positrons trapped, portable antimatter traps, and cold antimatter beams (e.g., with energy spreads ⩽1 meV) for precision studies of positron-matter interactions.