Antihydrogen, the simplest pure-antimatter atomic system, holds the promise of direct tests of matter-antimatter equivalence and CPT invariance, two of the outstanding unanswered questions in modern physics. Antihydrogen is now routinely produced in charged-particle traps through the combination of plasmas of antiprotons and positrons, but the atoms escape and are destroyed in a minuscule fraction of a second. The focus of this work is the production of a sample of cold antihydrogen atoms in a magnetic atom trap. This poses an extreme challenge, because the state-of-the-art atom traps are only approximately 0.5 K deep for ground-state antihydrogen atoms, much shallower than the energies of particles stored in the plasmas. [...]