The present disclosure provides a multi-stage compressor and an air conditioner having the same. The multi-stage compressor includes: a first-stage cylinder including a first-stage compression cavity and a first vane disposed in the first-stage compression cavity; a second-stage cylinder including a second-stage compression cavity and a second vane disposed in the second-stage compression cavity, wherein a refrigerant flowing out from the first-stage compression cavity enters the second-stage compression cavity; a linkage structure disposed between the first vane and the second vane, so that the second vane is capable of moving with a movement of the first vane and maintain contact with a roller in the second-stage compression cavity.