Sustained advances in the trapping of positrons and antiprotons led to the recent creation of cold antihydrogen in vacuum under controlled conditions. This was achieved at the unique Antiproton Decelerator facility located at CERN, Geneva. The collection, manipulation and mixing of clouds of the antiparticles necessary to promote antihydrogen formation are described herein, including some of the more practical aspects of positron accumulation. This discussion is prefaced by a treatment of basic Penning trap and plasma physics of relevance to antihydrogen formation. The detection of the nascent antihydrogen atoms, both via their annihilation on Penning trap electrodes and following field ionization of weakly bound pairs, is reviewed. We present a brief description of aspects of the physics output of the antihydrogen experiments in terms of the nature of the states which are formed and implications of measurements of the spatial distribution of antihydrogen annihilation events. Theoretical simulations of antihydrogen formation have been useful in providing guidance in interpreting experimental data, and aspects of this work are reviewed. Trapping of neutral systems using a magnetic-field minimum device is described and the new ALPHA antihydrogen trapping experiment is introduced. We conclude with a look to the future of the new field of antihydrogen physics.