Photon (X-ray) radiotherapy is the most common treatment used in cancer therapy. However, the exposure of normal tissues and organs at risk to ionising radiation often results in a significant incidence of low-grade adverse side effects, whilst high-grade toxicities also occur at concerningly high rates. As an alternative, boron neutron capture therapy (BNCT) aims to create densely ionising helium and lithium ions directly within cancer cells, thus sparing the surrounding normal cells and tissues but also leading to significantly more effective tumour control than X-rays. Although very promising for patients with recurring and highly invasive tumours, BNCT does not currently have widespread use worldwide, in part due to limited and reliable neutron sources for clinical use. Another limitation is devising strategies leading to the selective and optimal accumulation of boron within the cancer cells. Boronophenylalanine (BPA) is currently the major compound used in BNCT which takes advantage of the amino acid transporter LAT1 that is overexpressed in a number of human cancers. Additionally, there is a lack of in-depth knowledge regarding the impact of BNCT on cellular DNA, and the molecular mechanisms that are responsive to the treatment, which are important in developing optimal therapeutic strategies using BNCT, are unclear. In this review, we highlight the current knowledge of the radiobiology of BNCT acquired from in vitro and in vivo studies, particularly in the context of DNA damage and repair, but also present evidence of established and new boron-containing compounds aimed at enhancing the specificity and effectiveness of the treatment.