Metallic glasses (MGs) constitute an emerging class of advanced structural materials due to their excellent mechanical properties. However, brittle failure at room temperature and the resultant complicated fracture behavior greatly limit their wide engineering applications. Over the past decades, the deformation and fracture in ductile or brittle mode referring to material compositions, load conditions, sample size, etc., have been widely studied, and significant progress has been made in understanding the failure behavior of MGs. Micromechanisms of fracture have been revealed involving shear banding, cavitation and the nature of the crack tip field. The ductile-to-brittle transition and inherent governing parameters have been found. To well describe and predict the failure behavior of MGs, failure criteria for ductile and brittle MGs have been established empirically or based on atomic interactions. In this paper, we provide a detailed review of the above advances and identify outstanding issues in the failure of MGs that need to be further clarified.