Tools and methods for abrasively cutting downhole pipes and casing. The tools and methods are ideally suited for situations where the pipe to be cut or perforated is positioned partly or wholly inside another pipe and damage to the outer pipe must be avoided. The jet nozzles are positioned at a non-normal angle to the target surface to reduce the jets' effective cutting distance. While pumping the abrasive fluid, the jets are supported at a selected radial distance from the target surface so that within a predetermined operating time the jets will cut or perforate the inner pipe but leave the outer pipe substantially intact. The tool may be rotated with a motor to perform cutoff operations or held in a fixed position for perforating.