An abrasive deburring machine comprised of a plurality of individually controlled abrasive modules, each having a rotatable drum type brush with flexible bristles, the drum is mounted on a rotatable support with axis of rotation generally perpendicular to the rotational axis of the drum. Each support with brush and support drive are mounted inside of a housing therefore creating a module. Plurality of such modules can be combined in a desired configuration to form a deburring machine or finishing/polishing machine tailored to a specific geometry of the work piece or part to be deburred, finished or polished. Each drum and drum support, including its axial angular position are controlled independently, which for a pre-programmed movement of all brushes in order to conform to a complex shape of the part to be deburred, optimize the process of deburring and maximize the durability of abrasive brushes. The drum type brush is preferably made of polystyrene foam with helical groove along its outer shape containing a continuous abrasive filament secured inside the groove and having bristle or strap like abrasive extremities. The invention includes a variety of modes of use being mostly applicable for deburring and finishing operations in Aerospace sector as well as in finishing operations in other metal, plastic and woodworking industries.