High precision antihydrogen experiments allow tests of fundamental theoretical descriptions of nature. These experiments are performed with the ALPHA apparatus, where ultra-low energy antihydrogen is produced and confined in a magnetic trap. Antihydrogen spectroscopy is of primary interest for precision tests of CPT invariance - one of the most important symmetries of the Standard Model. In particular, the 1S-2S transition frequency in hydrogen is the most precisely known quantity in Physics thus measuring the same quantity with antihydrogen provides the most stringent comparison between matter and antimatter. Antimatter gravity is an open experimental question that deserves to be directly addressed in order to test the foundation of the General Theory of Relativity. Methods to produce, trap, detect and identify antihydrogen are presented in this work, alongside the first high precision measurement of an antihydrogen property, i.e., the electric neutrality of an antiatom. This measurement also constitutes a three-fold improvement to the measured value of the positron charge. The focus is then shifted to the proposed experiment to measure the antihydrogen gravitational acceleration, with particular attention to the antihydrogen detector.