An air cleaner includes a case, an air inlet, an air outlet, a grill covering the air inlet to filter airborne oil droplets, a fan to suction air through the air inlet and discharge air out of the air outlet, a roll filter provided between the grill and the fan to filter oil mist, and a filter roller to cut the roll filter. The roll filter includes a roller and a filter sheet wound around the roller, and the filter sheet may be unwound to extend behind the grill. When the filter sheet becomes dirty, the roll filter is further unwound so that a clean section is provided behind the grill, and the filter cutter may cut off the dirty section. During cutting, a clamp holds down the filter sheet. The case includes a slit through which the dirty section of the filter sheet passes for discarding.