This paper provides a comprehensive overview of how fitting of baryon acoustic oscillations (BAO) is carried out within the upcoming Dark Energy Spectroscopic Instrument’s (DESI) 2024 results using its DR1 data set, and the associated systematic error budget from theory and modelling of the BAO. We derive new results showing how non-linearities in the clustering of galaxies can cause potential biases in measurements of the isotropic (αiso) and anisotropic (αap) BAO distance scales, and how these can be effectively removed with an appropriate choice of reconstruction algorithm. We then demonstrate how theory leads to a clear choice for how to model the BAO and develop, implement, and validate a new model for the remaining smooth-broad-band (i.e. without BAO) component of the galaxy clustering. Finally, we explore the impact of all remaining modelling choices on the BAO constraints from DESI using a suite of high-precision simulations, arriving at a set of best practices for DESI BAO fits, and an associated theory and modelling systematic error. Overall, our results demonstrate the remarkable robustness of the BAO to all our modelling choices and motivate a combined theory and modelling systematic error contribution to the post-reconstruction DESI BAO measurements of no more than 0.1 per cent (0.2 per cent) for its isotropic (anisotropic) distance measurements. We expect the theory and best practices laid out to here to be applicable to other BAO experiments in the era of DESI and beyond.