The ASACUSA collaboration at the Antiproton Decelerator of CERN aims at a precise measurement of the antihydrogen ground-state hyperfine structure as a test of the fundamental CPT symmetry. A beam of antihydrogen atoms is formed in a CUSP trap, undergoes Rabi-type spectroscopy and is detected downstream in a dedicated antihydrogen detector. In parallel measurements using a polarized hydrogen beam are being performed to commission the spectroscopy apparatus and to perform measurements of parameters of the Standard Model Extension (SME). The current status of antihydrogen spectroscopy is reviewed and progress of ASACUSA is presented.