Cold atoms of antihydrogen promise a unique opportunity to study the properties of atomic antimatter, and via comparisons with its well-studied matter-counterpart, the possibility to test CPT invariance. This symmetry is conserved in local quantum field theories, so tests in varied systems provides the experimental validation of this framework. In order to probe matter-antimatter symmetry at the highest possible precision, it is essential that the anti-atoms be suspended in vacuum to allow for detailed interrogation via laser light or microwaves. The ALPHA experiment, running at the CERN antimatter factory, is now trapping sufficient numbers of antihydrogen atoms to enable these studies.
The best measurements in hydrogen are of the 1S-2S transitions (precision 0.01 ppt, determining the Rydberg),   the ground state hyperfine interval (precision 1ppt ), and the 2S-2P Lamb shift.  The trap environment is particularly challenging for spectroscopy and requires adaptation of the usual AMO techniques. I will present our recent measurements of these transitions in antihydrogen, where we have recently reported  2 ppt precision 1S-2S  and 9 ppm precision  ground state hyperfine interval results. Our current measurements of the antihydrogen 1s-2p transitions and demonstration of laser cooling together with the improved  antiproton beams from  ELENA  promise excellent prospects for even higher precision results.