The decision by the International Thermonuclear Experimental Reactor to use boron coated tungsten walls puts an emphasis on the spectroscopy of boron hydride, Boron hydride (BH), whose emissions can be used to monitor wall erosion. An accurate spectroscopic model for all six isotopologues of BH (BH, BH, BD, BD, BT and BT) is constructed and used to compute line lists for these species. These line lists, named AX, can be used to simulate both the A —X system in the 400 nm region and infrared transitions within the X ground. A MARVEL (Measured Active Rotation Vibration Energy Levels) study of observed BH high resolution spectra produces a list of 204 empirical energy levels which are used to tune potential energy curves used in the model, which results in root-mean-square errors between the calculated and empirical energy levels of 0.037 cm for the A state and 0.025 cm for the X state. The revised value between these states is 23151.78 cm. The MARVEL energy levels are also used explicitly to improve the energy levels and transition wavenumbers. Comparisons of our calculations with measured lifetimes and observed spectra give excellent agreement for BH. Our model is found to be good but less accurate for other isotopologues, but further improvements will require input from laboratory measurements. The line lists are made freely available on the ExoMol website (www.exomol.com).