Recent advances in techniques to manipulate and study, with high precision, atomic hydrogen, from one hand, and successful trapping schemes for positrons and antiprotons, from the other hand, have encouraged the pursuit of experiments to test CPT violation and the Weak Equivalence Principle (WEP) through the comparison of hydrogen and antihydrogen. A description of the hydrogen trap and laser system being built in Rio, to trap and perform high resolution spectroscopy on cold hydrogen, is presented along with a discussion on the techniques and experimental system being implemented by the ATHENA collaboration at CERN to produce cold antihydrogen. A new technique to make a cold antihydrogen beam is proposed.